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BY THE SAME AUTHOR. 

The Steam Engfine, and Gas and Oil 
Engines. 8vo. 7s. 6d. net. 



British Association Meeting in South 

Africa, 1905. Discussion at Johannesburg 
on the Teaching of Elementary Mechanics. 
Edited by Professor John Perry. 8vo. 
2s. net. 



LONDON: MACMILLAN & CO., LTD. 



ELEMENTARY PRACTICAL MATHEMATICS 



MACMILLAN AND CO., Limited 

LONDON • BOMBAY • CALCUTTA 
MELBOURNE 

THE MACMILLAN COMPANY 

NEW YORK • BOSTON • CHICAGO 
DALLAS • SAN FRANCISCO 

THE MACMILLAN CO. OF CANADA, Ltd. 

TORONTO 



ELEMENTARY 
PEACTICAL MATHEMATICS 



WITH NUMEROUS EXERCISES FOR THE USE OF STUDENTS 

AND ESPECIALLY OF MECHANICAL AND ELECTRICAL 

ENGINEERING STUDENTS 



BY 

JOHN PERRY 

M.E. D.Sc. LL.D. F.R.S. 



PROFESSOK OF MECHANICS AND MATHEMATICS AT THE ROYAL COLLEGE OF SCIENCE, LONDON 

PAST PRESIDENT OF THE PHYSICAL SOCIETY 

PAST PRESIDENT OF THE INSTITUTION OF ELECTRICAL ENGINEERS 



MACMILLAN AND CO., LIMITED 

ST. MARTIN'S STREET, LONDON 

1913 




^ 



K 



COPYRIGHT 



CONTENTS 



CHAPTER 



PAas 



I. Arithmetic . - - - 1 

II. Logarithms - - 8 

III. The Slide Rule - - - - - - ^ - - 15 

IV. Evaluation of Formulae .1 J . . - - 20 
V. Algebra . _ - 25 

VI. Mensuration 51 

VII. Angles 61 

VIII. Speed 68 

IX. Uses of Squared Paper 79 

X. Some Mensuration Exercises ----- 88 

XI. Mensuration Exercises 93 

XII. Squared Paper ...--.-- 99 

XIII. The Linear Law - - -. 106 

XIV. Squared Paper 115 

XV. Important Curves - - - - - - - 126 

XVI. Squared Paper - - - 132 

XVII. Maxima and Minima 136 

XVIII. The Infinitesimal Calculus - - - - - 139 

XIX. Formula and Proofs 149 

XX. The Calculus - - - - - - - - 151 



vi CONTENTS 

CHAPTER PAGE 

XXI. Illustrations 163 

XXII. Maxima and Minima 168 

XXIII. Curves 174 

XXIV. Illustrations 177 

XXV. Illustrations. Beams and Struts - - - 181 

XXVI. Illustrations. Fluid 184 

XXVII. The Compound Interest Law - - - - - 189 

XXVIIT. Simple Vibration 213 

XXIX. Mainly about Natural Vibrations - - - 224 

XXX. Forced Vibrations 234 

XXXI. Periodic Functions in General - . . . 244 

XXXII. Extended Eules and Proofs 249' 

XXXIII. Exercises with Unreal Quantities - - - 263; 

XXXIV. Fundamental Equations 274 

XXXV. Telephone and Telegraph Problems - - - 277 

XXXVI. Heat Problems 290 

XXXVII. Vectors 294 

Board of Education Examinations . - - 308 

Index 33a 



INTRODUCTION 

Academic methods of teaching Mathematics succeed with about 
five per cent, of all students, the small minority who are fond of 
abstract reasoning : they fail altogether with the average student. 
Mathematical study may be made of great value to the average 
man if only it is made interesting to him. The name Practical 
Mathematics has been given to a new method of study, not because 
it describes the method but merely to differentiate it from the older 
method. 

I began to use the new method forty years ago in an English 
public school, later in Japan. In 1881 I was bold enough to make 
it part of the curriculum at the City Guilds Technical College, 
Finsbury. It proved so successful that I induced the Board of 
Education to make it part of their scheme for Science classes. The 
number of Science class students of Practical Mathematics in 
Great Britain increases at a greater rate than the Compound 
Interest law, and at present there are more students of this subject 
than of any other. If a class is formed in September in elementary 
Pure Mathematics (Pure Mathematics is the name given to the 
older or academic method of study) and twenty students join it, 
at Christmas the number has dwindled to seven, and only one or 
two keep in attendance till May, whereas a class in Practical 
Mathematics keeps up its attendance almost intact to the end of 
the session. 

There is always a difficulty in obtaining competent teachers. 
Any man who has learnt Pure Mathematics is thought by himself 
and others to be fit to teach, whereas his very fondness for and his 
fitness to study Pure Mathematics make it difficult for him to 
understand the simple principles underlying the new method. 



viii INTRODUCTION 

We show a student how to work problems, exercising his common 
sense, and we give him experimental proof of the correctness of his 
results. Our methods of reasoning are those logical methods which 
are adopted in the teaching of Physics and in common affairs, and 
we claim that he understands and takes an interest in what he is 
doing, and feels confidence in his results. 

When the Board established the new subject in 1899, 1 was asked 
to give a course of six lectures on Practical Mathematics to working 
men in London. A summary of these lectures, illustrated by 
exercises, was published by the Government. This has been re- 
published by the Government with very large additions to the 
exercises, nearly all the important questions in the examinations 
from 1900 to 1909 being incorporated, as they are also in this book. 
It is found that, in spite of the cheapness of this publication, teachers 
do not consult it, nor do they seem to consult either the examina- 
tion questions or the yearly reports of the examiners which are also 
published. 

A great number of text^books of Practical Mathematics based 
upon the above summary have been published since 1900, and some 
of them are good, containing many excellent exercises like those 
I have mentioned ; but, in spite of the guidance of these text-books, 
it is well known that there are many thousands of earnest hard- 
working students who are n the hands of teachers who do not 
understand our methods. They make their students work many 
text-book examples, but there is no real teaching. The best remedy 
is the doubling of salaries of the teachers. 

The Board occasionally gives to a limited number of teachers 
the opportunity of coming to London to attend a three-weeks' 
course of instruction in July. Invariably I find that these teachers 
are the most earnest and hard working of students. I shall try 
in the following pages to reproduce part of the information which 
I give them, as well as the exercises which they are asked to gixe 
to their students, and to work themselves. I have usually one 
assistant for every ten students, his duty being that of helping in 
the exercise work. 

I usually begin by directing the attention of students to some of 
my own published addresses, copies of which may be borrowed, and 
I make some introductory remarks which are merely amplifications 
of the following statements. 



INTRODUCTION ix 

Essential things are neglected in our methods of primary and 
secondary education. Teachers have been brought up on bad old 
systems unsuited to modern needs. There are too many pupils 
per teacher, and teachers are badly paid. Useful reform is taking 
place in primary schools, but in secondary schools we still teach 
as if all average boys were proceeding to classical studies at the 
University, and preparing to be clergymen or schoolmasters. 

The average English boy remains uneducated except through his 
sports ; he learns nothing in the class rooms. There is hardly any 
business in which it is not essential to have a knowledge of English, 
a knowledge of computation, and a training in Natural Science, 
and success is hardly possible for any man who does not love books. 
It is easy to give these qualifications to any English boy, yet he 
almost never gets them. Again, the most prominent Englishmen 
understand nothing of those sciences which are transforming all the 
conditions of civilisation. We adhere to medieval ideals. 

Five hundred years ago all books were in Latin. A man could 
not read unless he knew Latin ; now, English Literature (including 
translations), is greater than any other ever known; but we still 
assume that a man is illiterate if he does not know Latin. Five 
hundred years ago only one man in a hundred could write his 
name ; hardly anybody could compute. We have discarded some 
old methods of teaching, and now every boy in the country can read 
and write, and he can perform arithmetical work which was quite 
beyond the powers of the greatest Alexandrian philosophers. No 
one dreams of philosophising over Euclid's 7th, 8th, 9th, or 10th 
books, and in the same way all the books of Euclid ought to be 
superseded. For much of the 2nd and 5th books of Euclid we need 
only a page of easy algebra. After a man has become a Cambridge 
wrangler he will find the fifth book of Euclid, and indeed the whole 
of Euclid, a most fascinating study, but it is a cruel jest to call 
a boy stupid because he finds the study impossible. 

The average boy cannot take to abstract reasoning, and he is 
called stupid ; I think him much wiser than the boy who is usually 
called clever. He ought to be actually acquainted with concrete 
things before he is asked to reason about them. Children should in 
their play be accustomed to measurement ; playing at keeping 
shop, selling things to each other by weight and measurement, 
paying for things in actual money. 



X INTRODUCTION 

Measurement of things with callipers and scales would accustom 
boys of eight to the use of decimals. Boys of ten will sketch and 
draw plans of their schoolhouse and the roads or streets about it ; 
they soon learn to use maps, and they know that maps may be 
of different scales. They ought to be led up to the vector subject 
of Geometry only slowly, through maps useful to themselves, 
through problems on heights and distances and the like, and almost 
intuitively they would get an exact knowledge of the subjects of the 
sixth book of Euclid. Easy Mensuration is a fascinating subject to 
the average boy, and cultivates his reasoning powers if he measures 
and computes and tests his answers by experiment. I consider that 
no boy can get mental training through any subject of study unless 
he is interested in it and happy, and therefore I think that a boy's 
main business at school is to continue the study of observational and 
experimental science to which he has been accustomed since the 
day he was born. The methods of the ordinary schoolmaster are 
all opposed to this idea, and the worst of these methods is the 
teaching of many subjects in water-tight compartments, instead of 
teaching many subjects incidentally in the teaching of some one 
subject. 

We make two very great mistakes : we painfully give a child 
the impression that an idea is difficult to understand, an idea 
perfectly familiar to it since it was three years of age, and again we 
assume that a child can understand quite readily some grown-up 
idea that looks to us simple. Thus we produce stupefaction, and 
the child is mentally shipwrecked; it seizes upon any raft for 
safety, and the raft ever ready is a formula, a rule. If school- 
masters studied their pupils as trainers of animals do, they would 
find the average boy capal^le of the highest kinds of intellectual 
work. 

After a boy has made up by the use of objects, a multiplication 
table and can multiply and divide, I feel sure that he ought to 
leave mere number and get to quantity, so that he may use decimals 
quite early. To teach him the use of decimals and to educate his 
hand and eye and judgment, you allow him to measure things. 
But how do you do it 1 Often in the most uninteresting way ! He 
is made to measure a certain length, to weigh an object, etc., and 
his answers are compared with the real length or weight, etc. 
Contrast this with the following exercise. He is given a block of 



INTRODUCTION xi 

iron, and he measures its length, 3-27 inches; breadth, 2-63 inches; 
thickness, 1*95 inches. He finds its volume to be 16*77 cubic 
inches. He readily uses contracted methods. Why will you try 
to stop him 1 He is given a cube 1 inch in edge of the same kind of 
iron. He takes it to the scales, and finds that it weighs 0*26 lb., so 
he computes the weight of his block to be 4*36 lb. He now goes 
and weighs it, and is delighted. Do you see where the difference 
comes in, and how interesting it is to find his computation agreeing 
with reality? And that same piece of iron is allowed to displace 
water in a vessel, and the displaced volume is found experi- 
mentally to be nearly 16-77 cubic inches. He now takes an 
irregularly shaped piece of iron, finds its volume by displacement, 
computes its weight, and weighs it by the scales. He sees at once 
how it is that perfect agreement cannot be expected. 

Common-sense explanation accompanying experiment ought to be 
the rule. Do not teach abstract geometry at all; teach mensura- 
tion with the help of arithmetic and algebra and weighing and 
measuring. Do not be afraid to introduce a boy early to sines, 
cosines, and tangents, and the calculation and actual measurement 
by surveying, etc., of heights and distances. Boys are intensely 
Interested in such work, and it is educational. But how awfully 
dull it may be made ! 

Besides this algebraic side of Practical Mathematics there is 
Graphics, which comprehends practical plane and solid geometry, 
and the summation of vectors and forces. 



NOTE 



If a sufficient number of the teachers attending a summer course 
desire to have some advanced instruction, I usually give them one 
lecture each day. They are expected to do most of the exercises 
given to the others, but special advanced exercises are also given. 
As such advanced work must take only one of several possible 
directions, I usually ask these students to choose the direction. 
This year, 1912, they chose the subject of harmonic functions, and 
the exercises which concern vibrating bodies, alternating currents 
of electricity, submarine telegraph and telephone circuits, heat 
conduction, etc., are given towards the end of this book. These 
' advanced ' exercises can all be readily worked by any student who 
faithfully works through the earlier or elementary ones. A student 
may think that they are of practical importance only to electricians ; 
but it would be easy to set exactly the same mathematical exercises 
as mechanical engineering or general physics problems without 
introducing one technical term of the electrician. 

The Board of Education examines now in two stages only. Can- 
didates for the lower examination ought to be acquainted with 
almost all the work of Chapters I. to XVIII. of this book. Can- 
didates for the higlier examination ought to be acquainted with the 
whole of the book. I have given the answers to the papers set for 
the years 1910, 1911, and 1912. 

The vile system of examinations created by such institutions as 
London University makes it to the interest of a student to attend 
the classes of a cheap crammer rather than those of a real teacher ; 
it has also created that kind of text-book which exactly suits one 
examination and no other. The subject of Practical Mathematics is, 
I am happy to say, a subject which is not likely to commend itself 



xiv NOTE 

to such institutions, nor are such text-books likely to be of much 
use to real students. This book is primarily for teachers, but it 
seems to me that a book so different from a cram text-book is the 
best of all text-books for a student. 

Few people comprehend the difficulty of setting examples which 
are really practical. I have, myself, constructed all the exercises 
and questions given in this book. In many cases each of these 
questions has taken several hours, and in some cases several days, 
to construct. 




ELEMENTARY 
PRACTICAL MATHEMATICS. 

CHAPTER I. 

ARITHMETIC. 

1. When calculating from observed quantities it is dishonest 
to use more figures than we are sure of. A boy who experiments 
learns this very quickly. 

Exercise. A boy measured a block of iron to be 6*56 inches long, 
4-13 inches broad, and 2*67 inches thick. He took a cubic inch of 
what was presumably the same kind of iron, and he found it to 
weigh 0'26 lb. Therefore the weight of his block in pounds ought 
to be 6-56 X 4-13 X 2-67 X 0-26. Multiplying out, he obtained the 
answer 18-80763976. 

Another boy measured the same block with the same measuring 
instruments, and found the length, breadth, and thickness to be 6*55, 
4'14, and 2*68. He found the weight of the cubic inch of iron to 
be 0*26 lb. His answer is therefore 

6-55 X 4-14 X 2-68 x 0-26 = 18'895... . 

The two answers differ because, with the instruments employed, 
there was a possibility of error in the last figure of each of the 
measurements, and it is evident that the first boy ought to say that 
his answer is 18*8 and the second boy 18'9, because, even when he 
only gives three significant figures in his answer, there is a possi- 
bility that the last figure is in error. Of course all figures after the 
first three are worthless. 

In a leading newspaper a few days ago I saw the indicated horse- 
power of a marine engine quoted as 3562 '74 horse-power. Well, it 
is very probable that this measurement is in error at least 5 per 
P.M. - A e 



2 ELEMENTARY PRACTICAL MATHEMATICS 

cent. That is, the person who made the measurements and calcula- 
tions is not sure whether the answer might not be 3700 or 3400, 
and yet he pretends that his last iigure 4 has a meaning. I am 
sorry to say that many misleading figures of this kind are published 
in the best books written on the steam engine. 

Sailors being examined for their Mate Certificates calculate their 
traverse tables to seconds ; but their leeway is estimated in points ; 
the error of several degrees is almost certain to occur. They use 
six or seven-figure logarithms in their work ; four-figure logarithms 
are sufficieutly accurate. In the examinations of the Institution 
of Civil Engineers, although only three figures are needed in the 
answers, it is imperatively required that seven-figure logarithms 
should be used. 

When I was at school the mean distance from the earth to the 
sun was stated as 95,142,357 miles. I wonder why furlongs and 
inches were not mentioned. The best knowledge we now have of 
this distance is that it is not greater than 93 nor less than 92 J 
millions of miles. 

2. It will greatly prevent this sort of error if students will get 
into the way of writing 9*3 x 10^ or 9-25 x 10^, instead of 93,000,000 
or 92,500,000. This is convenient in other ways. 

In the following table note that we count how many places the 
decimal point must be moved to convert 6-548 into the number 
in question. 



654,800,000,000 
654,800 
65-48 

6-548 

0-6548 

0-006548 



6-548 X 10" 
6-548 X 105 
6-548 X 101 
6-548 X 100 
6-548 X 10-1 
6-548 X 10-3 



When I calculate I seldom trouble my head about the position 
of the decimal point in my answer until everything else is finished. 
There are many cleverly contrived rules about the position of the 
decimal point, but we forget them in practical work. Better never 
learn them. 

3. Multiplication. In multiplying 2-714 x 15-68, let us first 
neglect the positions of the decimal points and multiply 2714 by 1568. 
Let us suppose that we only want four significant figures in the 
answer. I here give the ordinary method; then I put for the 



ARITHMETIC 3 

unnecessary figures; then I show how I actually work. Let a 
student study, understand, and practise for himself. 

2 714 2714 2714 

.1568 1568 8651 



211712 


22000 


2714 


162184 


16300 


1357 


135710 


13570 


163 


27141 


2714 


22 



42551552 4256000 4256 

I write the multiplier backwards. In multiplying by the 6, say, 
I am supposed to multiply only on the 27, the figure above the 6 
and all to the left of it ; but I add 1 because 6x1 = 6, and I cannot 
reject this as it is nearer to 10 than to 0, so I carry 1. Students 
must practise and get very familiar with this shortened method 
of multiplication. Some men prefer not to write the multiplier 
backwards, their work, however, being what I have given. Now, as 
to the position of the decimal point, we are multiplying a number 
which is between 2 and 3 by a number which is nearly 16, so that 
42*56 is obviously our answer and not 4-256 or 425-6. 

4. Division. Divide 2714 by 1568, giving only four significant 

figures in the answer. Here is the ordinary method : all the figures to 

the right of the dotted line and below AB are unnecessary ; hence the 

merit of the other method, in which we cut off" figures in the divisor. 

1568)2714 1 173086 100^)2714 1 1731" 



■X- 



1568 


1568 


1146iO 
1097:6 


1146 
1098 


48:40 
47104 


48 
47 


13600 
,l;2544, 

110560 
9408 


1 



1152 
There is a further contraction of work, the above numbers 1568, 
1098, and 47 not being written, which ought to be taught to boys. 
I give the above because I use it myself. 

* Instead of cutting off figures in the divisor, some people write the answer 
backwards] under the divisor like this lo'-i- 



ELEMENTARY PRACTICAL MATHEMATICS 



If you want to have no doubt about the correctness of your 
fourth figure in either of the above cases of multiplication or 
division, you had better use 27140 instead of 2714 ; in fact, get an 
answer with five significant figures and reject the last of them. 

Ex. 1. Multiply the following numbers a and h, and also divide. 
I give the answers. 



a 


b 


Answers. 




ab 


alb 


bla 


1323 
17-56 
0-5642 
4-3-26 
0-01584 


24-32 
143-5 
0-2471 
003457 
2 104 


321^5 
2520 
0-1394 
0-01495 
0-03333 


54-4 
•12-24 
2-283 
1251-0 
0007530 


0-01838 
8-172 
0-4380 
0-0007991 
132-8 



Ex. 2. To find the Napierian logarithm of a number n (this is 
called loggTi), we multiply the common logarithm (this is called 
log^o^) V '-^'3026. Convert the following : 



logioft 


Answers. 


loggn 


2-1469 

0-3574 

-1-5178 

-3-2005 


4-9435 

0-8-2-29 

-3-4949 

-7-3695 



When given logg^i we divide it by 2*3026, or, what is the same 
thing, we multiply it by 0*4343 to find the common logarithm. 

Ex. 3. Convert the following : 





Answers. 


logeW 


Answers. 




logion 


logion 


5-7152 

0-3513 

-2-1435 

-4-7354 


2-4821 
0-15-26 

- 0-9309 

- 2*0566 


-2-1543 
1-7216 
8-4175 

- 0-1493 


-0-9256 
0-7477 
3-6557 

- 0-0648 



Let the student make himself quite sure that dividing by 2-3026 
is really the same as multiplying by 0-4343. 

5. Percentages. When we say 5 per cent., we mean 5 per hun- 
dred or 5-f 100 or 0*05. Thus 0*035 means 3J per cent., 0-02 
means 2 per cent. 



ARITHMETIC 5 

Some quite wrong expressions are in such common use that we 
do not condemn them. For example, a broker says he will charge 
you half-a-crown per cent. He means half-a-crown per hundred 
pounds : this is really ^th of one per cent., whereas his language 
implies half-a-crown per hundred half-crowns, which is one per cent. 
One cow per cent, means one cow per hundred cows, or one per 
cent. But we forgive the broker. 

Ex. 1. A piece of alloy weighing 3"28 lb. contains 2*65 lb. of 
copper, 0*46 lb. of tin, 0'17 lb. of lead; state these as percentages 
of the whole. Answer: 2-65 -^ 3-28 = 0-8079 or 80-79 per cent, of 
copper; 0-46 -r 3-28 = '1402 or 14-02 per cent, of tin; 0-17 -r 3-28 
= -0518 or 5 '18 per cent, of lead. 

Ex. 2. By measurement and computation two men find the 
weight of a body to be 15-72 and 15*59 tons. Assuming the mean 
of these two figures to be right, what is the percentage error in 
each"? 

The mean is J(15-72 -f 15*59) or 15-655; each differs from this 
by 0-065; the fractional error is 0-065 -r 15*655 = -00415 or 0*415 
per cent. 

Ex. 3. A man whose income is 230 pounds per annum has it 
increased by 5 per cent. What is the new income *? Answer : 
add -05x230 or 11*5; it is 241*5. Or we might have multiplied 
230 by 1-05. 

Ex. 4. A man's income is 241*5 pounds; it is reduced to 230; 
what is the percentage reduction 1 Answer: 11-5-^241-5 = -0476 
or 4-76 per cent. 

Ex. 5. I can buy £100 of 4 per cent, stock for £86. What 
income do I obtain ? Answer : for each £86 I get £4 income ; that 
is the fraction 4 -f- 86 or 0-0465 or 4-65 per cent, of my capital. 

6. Commercial Arithmetic. As our sums of money, etc., are not 
expressed in a decimal system, before we can multiply or divide we 
find it necessary to express them in the decimal system. Bank 
clerks, grocers, and others who have many calculations to make of 
the same kind use labour-saving rules to effect such objects. When 
a grown-up student s6es what the object is he easily understands 
such rules, but he need not hamper his memory with such rules 
unless they belong to his trade. The principle is easily understood 
from the following exercises. 

1. Convert £182. 175. ^d. to the decimal system. 

9 17-75 

Here 9t?. is — or 0-755. ; we have therefore 17-75s. or — ~ or 
12 20 

0-8875 pounds. Our answer is therefore 182*8875 pounds. 



6 ELEMENTARY PRACTICAL MATHEMATICS 

2. Convert 3 tons 5 cwt. 2 qrs. 18 lb. to the decimal system. 

18 „_ 2-643 „„.,_ 5-6608 „^_, 
28 " ^^^ '' ~T~ "" ^ ~20~ "^ ^ 

so that the answer is 3-28304 tons. 

3. Convert 5 miles 3 furlongs 30 perches 4 yards to the decimal 
system. 

4 8 ^^ 30-73 ^^^ 3-768 ,^, 

so that the answer is 5-471 miles. 

The converse process will be understood from the following 
examples. 

4. Convert 7*828 pounds into pounds, shillings, and pence. 

-828 pounds = -828 X 20 shillings or 16-56, and -56 shillings = 
-56 X 12 pence or 6-72, so that the answer is £1. 16s. 6-72c?., or 
£1. 16s. ^d. nearly. 

5. Convert 56-2154 tons into the usual absurd English form. 
•2154 tons = -2154 x 20 cwt. or 4-308, and -308 cwt. = -308 x 4 qrs. 

or 1-232; -232 qrs. = -232 x 28 lb. or 6^ lb., so that the answer is 
56 tons 4 cwt. 1 qr. 6J lb. 

6. Convert 3-78085 miles. Here -78085 miles = -78085 x 8 furlongs 
or 6-2468; -2468 furlongs = -2468 x 40 perches or 9-872; -872 
perches = -872x51 yards = 4-796 yards; 0-796 yards = 2388 feet; 
•388 feet = 4-656 inches; so that our absurd-looking answer is 
3 miles 6 furlongs 9 perches 4 yards 2 feet and 4-656 inches. 

This kind of absurdity is flaunted in our faces in school books as 
if it were admirable, and there is no man to say that it is wicked. 
Truly it is only a special providence that can have prevented the 
ruin of the English people. 

7. The rules called Practice, Interest, etc., are merely easy 
examples in multiplication and division in countries where decimal 
systems of money and weights and measures are employed. 

Ex. 1. What is the cost of 33-24 yards of ribbon at 7-86 cents 
(or 0-0786 dollars) per yard? Answer: 33-24x0*0786 or 2-613 
dollars. (This may be read 2 dollars 61 cents.) 

Ex. 2. What is the cost of 11-275 kilogrammes at 1-234 francs 
per kilo"? Answer: 11-275x1-234 or 13-913 francs. (This may 
be read 13 francs 91 centimes.) 

Interest. Exercise. The sum of 5143-65 dollars is lent for 
302 days at 4 J per cent, per annum. What is the interests Here 
the interest for 365 days is 5143-65 x ^045, so that this multiplied 
by 302 and divided by 365 is the answer, or 19r51 dollars. 



ARITHMETIC 7 

Proportion. If 3*275 tons cost 1-625 pounds, what will 1*164 
tons costi Here 1 ton costs 1*625 -=-3*275, and this multiplied by 
1*164 is the answer, or 0*578 pounds. 

Any person who examines an English Arithmetic will see that 
the troubles of boys are twenty times as great as they would be if 
we used decimal systems. And not only is this the case, but a boy 
who is expected to know the reasons for the rules he uses is intro- 
duced to complex logic which is far beyond his powers. I think 
that it is just here that a boy gives up all hope of being able to 
reason about such matters, and afterwards he refuses to make any 
serious effort to reason. 



CHAPTER 11. 
LOGARITHMS. 

8. The use of logarithms enables us to compute much more 
rapidly than by ordinary arithmetic. 

Many kinds of computation seem almost hopelessly difficult 
except by the use of logarithms. 

The symbol a^ means ax ax a. 

Hence 23 = 8; 2^ = 32. 

Many people say " a^ means a multiplied on itself three times." 
Of course this is wrong. It is, however, right to say, "a^ means 
1 multiplied by a three times." Thus a^ means 1 multiplied by a 
no times, or a^ is really 1. 

Definition of a Logarithm. 

If ft" = iV, then n = log„ A", and we read this as " ?i is the logarithm 
of N to the base a." Thus 2^ = 8, and hence 3 is the logarithm of 8 
to the base 2. 

We almost always use only logarithms to the base 10 in arith- 
metical work because we use the decimal system of writing num- 
bers ; but in many important calculations we need to use Napierian 
logarithms whose base is 2*71828, a number so important that the 
letter e is generally used to denote it, just as the Greek letter tt is 
used to denote 3*14159. It can be shown that if we multiply the 
common logarithm of any number by 2*30258 or divide by 0*43429 
we get its Napierian logarithm.* 

Whether n and m are integers or not, we define ft" to be such that 

ft"Xft"* = ft"'+". 

♦Note. — If e^ = N=W\ then x is the Napierian and n is the common 
logarithm of N ; \og-^QN=n = x\ogi(^e and logioe = 0*43429. Hence we divide 
n by 0*43429 to get x. 



LOGARITHMS 9 

It follows from this that 

a" -^ «"* = «"-'", 
and also that («")"* = a"'". 

Hence a"-^a" = ftO_l, 

and 1 ^ a'* = ao ^ a" = ci""" = a"". 

Let a student take any number which I shall call a; let him 
extract a very high root of it either by logarithms or by continued 
extraction of square roots ; he will find that the higher the root, 
that is the nearer he approaches to a^, the nearer does it get to L 
The student must not scorn an exercise like this ; he need not work 
it if he thinks he knows this thing well enough already, but it is by 
working such exercises that a man gets a real intimate acquaintance 
with the subject. 

We have then the rules : 

L Add the logarithms of two numbers and we have the logarithm 
of their product. 

2. Subtract the logarithms of two numbers and we have the 
logarithm of their quotient. 

Again, the logarithm of a^ is twice the logarithm of a. 

„ „ a^ is three times „ a. 

„ „ a^ or sfa is half „ a. 

„ „ a^ or sfa is one-third „ a. 

In the same way, if b is any number whatsoever the logarithm of 
a* is b times the logarithm of a. 

(In the first 18 chapters of this book, instead of writing the 
common logarithm of a number as log^jTi, I shall write it logn.) 

9. I have given each of you a copy of a set of tables of four- 
figure logarithms and anti-logarithms published by the Board of 
Education ; there is also a table of the sines, cosines, tangents, and 
radian measure of angles between 0° and 90°, and the first page of 
the little pamphlet gives various useful numbers and formulae. 
See the tables, Chap. VIII. 

Later, I shall tell you how to calculate a table of logarithms. 

^ow to use the table headed " logarithms," notice that there are 
no decimal points anywhere. Given the number 5204, we look at 
52 on the left, and above, this gives us 7160; our 4 causes us to 
look for the small number 3 on the right-hand columns, and we add 



10 ELEMENTARY PRACTICAL MATHEMATICS 

it and so get 7163. The smallest amount of practice is surely 
enough to teach this. 

Now I want you to understand that this means 
log 5-204 = 0-7163. 

You will see, therefore, that to find the logarithm of any number, 
it is only necessary to study the following illustrative examples. 
Multiplying a number by 10 adds 1 to its logarithm. 



Number. 


Number as written 
in Art. 2. 


The logarithm is 


The value of the 
logarithm is more 






compactly written 


520400',, 


5-204x105 


•7163 + 5 


5-7163 


5204 


5-204x103 


•7163 + 3 


37163 


5-204 


5-204x10° 


•7163 + 


0-7163 


•5204 


5-204 X 10-1 


-7163-1 


1-7163 


•005204 


5-204x10-3 


-7163-3 


3-7163 



The student notices that the whole number part of a logarithm 
depends on the position of the decimal point in the number. 

If you like, in 5-7163 or 3-7163 you may call the 5 or 3 (the 
whole number part) by such names as iTidex or characteristic, and 
the decimal part, which is . always positive, you may call the 
mantissa; but in truth you will do much better if you scorn the 
use of unnecessary technical terms like these. 

10. Again, given the logarithm of a number, to find the number. 
Proceed backwards as from column 4 to column 1 of last table. 

Thus, to find the number whose logarithm is 3-7163. In the 
antilogarithm table (we might do it with but little increase of 
trouble by means of the logarithm table itself), find what corre- 
sponds to -7163 ; it is evidently 5200 + 4 or 5204. This means that 

log 5^204= -7163, 
and hence log 5204 =3-7163. 

A little practice in multiplication and division will make you 
perfectly familiar with and give you a thorough understanding of 
this subject. Without such practice it is as useless to read minute 
instructions as it is to try to learn to swim or to ride a bicycle by 
reading instructions. 

The student ought now, using logarithms, to work the exercises 
given in Art. 4. 



LOGARITHMS 11 

Exercise. Find the common logarithms of the following numbers : 



Number. 


Logarithm. 


Number. 


Logarithm. 


7-135 


0-8534 


1-065 


0-0273 


713-5 


2-8534 


0-01065 


2-0273 


0-7135 


1-8534 


10-65 


1-0273 


0-007135 


3-8534 


106500 


5-0273 


71350 


4-8534 


1065 


3-0-273 



If we are asked to multiply or divide a logarithm itself it is 
necessary to remember that the whole number part is sometimes 
negative. 

Exercise. Find the values of 

V8574, n/8^574, s/MEU, v/O-0008574. 

The logarithms of these numbers have to be divided by 2, as the 
logarithm of a square root is just half the logarithm of the number. 



Number i^. 


logi^. 


ilogN. 


Answers. 


8574 
8-574 
0-8574 
0-0008574 


3-9332 
0-9332 
1-9332 
4-9332 


1-9666 
0-4666 
1-9666 
2-4666 


92-60 
2-928 
•9260 
•02928 



Thus to divide 1^9332 by 2, I remember that it is really 
- 1 + -9332 ; this is the same as - 2 + 1-9332, and the half of this 
is -l + ^9666 or 1-9666. 

To divide T-9332 by 3, I call it - 3 + 2^9332, so that the answer 
is -1+0-9777 or T-9777. 

To divide or multiply 3-9332 by such a number as 4-56, it is 
necessary to convert it all into the negative form. It is - 3 + 0^9332, 
and this is really - 2-0668._ Dividing this by 4-56 we get - 0*4532, 
and this is - 1+0-5468 or 1-5468. 

Thus (-00857)^ is 0-3522. 

I think that perhaps, when a logarithm like 2*4666 has to be 
divided or multiplied, the student ought at first always to put it, 
not as -2 + -4666 but as -1-5334, even when he multiplies or 
divides by 2 or 3. This may not be so quick as the other method, 
but the student knows better what he is doing, and is less likely to 
make mistakes. 



12 ELEMENTARY PRACTICAL MATHEMATICS 

EXERCISES. 

1. Calculate a*. That is, the number a raised to the power indicated 

Find the logarithm of a, multiply it by 6, and this is the logaiithm of 
the answer : 

Leta = 20'52 and 6 = 2. Ans. ^"iX'l. 

„ a= 1-564 „ b=n. „ 1-955. 

„ a= 0-5728 „ 6 = 3." „ 0-1879. 

„ a = 60-71 „ h = l. „ 3-930. 

Note here that to multiply by l^ means that we are to divide by 3. 

a=0-2415 and 6 = ^. Ans. 0-6227. 

«= 1-671 „ 6 = 2. „ 2-793. 

a= -5014 „ 6 = 3|. „ 0-08919. 

2. Without using logarithms, find 4-326 x 0-003457 to four significant 
figures, leaving out all unnecessary figures in the work. 

Find 0-01584^2-104 to four significant figures. 
Also do these, using logarithms. Find loge7. 

Calculate 6^--^\ 3-"^^% •042^-'''% 4^246^ 30-01^x0-026417 

A71S. 0-01495, 0-007529, 1-94592, 49-95, 0-7632, 0-2211, 3-008, 5-745. 

Teachers make a mistake when, in calculating a*, they regard a 

negative value of 6 as creating an exercise for advanced students 

only. The most elementary students ought to be accustomed to 

such exercises ; they will find no difficulty in them if their teachers 

do not introduce difficulties. 

11. EXERCISES ON CONTRACTED METHODS. 

1. Assuming that v^3 = l'7321 and \/2 = 1-4142, find by contracted 
methods, as accurately as the given numbers will allow, \/6 and \/l-5. 

Ans. 2-4495; 1-2248. 

2. Having given that to five significant figures \/5 = 2-2361 and 
'/2 = 1'4142, find by contracted methods v/lO and V2-5. 

Ans. 3-1623; 1-5812. 

3. Find by contracted methods correct to four significant figures, 
without using logarithms, 0-30103 x 0*026007. Aois. 0-007829. 

4. A rectangle measures 23-59 x 18*64 cms. Using contracted methods, 
find the area and the ratio of its length to its breadth. 

Am. 439-7 ; 1-265. 

5. Having given that e^ = 1*3956, find by contracted methods e^ and e~^. 

Ans. 1*9477 ; 0*7165. 

6. Having given that ^^3 = 1*4422 and tbat_\^2 = 1*2599, find by con- 
tracted methods, without using logarithms, ^^6 and ^1*6. 

Ajis. 1*8170; 1*1447. 

7. To five significant figures 10^ = 3*1623 and 10^ is 1*3335, find by 
contracted methods 10^ and loi Ans. 4*217; 2*3715. 



LOGARITHMS 



13 



8. The diameter of a circle is 27*35 inches. Using contracted multipli- 
cation, show that the area of the circle lies between 587 and 588 square 
inches. The area of a circle is the square of its diameter multiplied 
by 0-7854. 



9. If to five significant figures e = 2"7183, find e^ 
method. Ans. 7-3891. 



by a contracted 



12. EXERCISES USING LOGARITHMS. 

1. For the following values of x calculate otf' in the following cases : 









Answers. 


Values of a:*». 






frlVATl ValllPR 














olx. 


When n 


Whenn 


When 71 


When n 


Whenn 


When 71 




is 4. 


isl. 


isi. 


is -\. 


is -1. 


is -4. 














00 


oo 


00 


0-1 


0-0001 


0-1 


0-5623 


1-778 


10 


10,000 


0-2 


0-0016 


0-2 


0-6688 


1-495 


5 


625 


0-3 


0-0081 


0-3 


0-7401 


1-351 


3-333 


123-5 


0-4 


0-0256 


0-4 


0-7952 


1-257 


2-5 


3906 


0-5 


0-0625 


0-5 


0-8410 


1-190 


2 


16 


.0-6 


0-1296 


0-6 


0-8802 


1-136 


1-667 


7-711 


0-7 


0-2401 


0-7 


0-9147 


1093 


1-429 


4-165 


0-8 


0-4096 


0-8 


0-9457 


1057 


1-250 


2-443 


0-9 


0-6561 


0-9 


0-9740 


1-027 


1-111 


1-524 


1-0 


1-000 


10 


1-0 


1-0 


10 


1-0 


1-1 


1-4641 


1-1 


1-0-24 


0-976 


0-9091 


0-6830 


1-2 


20736 


1-2 


1-046 


0-955 


0-8333 


0-482 


1-3 


2-8561 


1-3 


1-068 


0-937 


0-7692 


0-3501 


1-4 


3-8416 


1-4 


1-088 


0-919 


0-7143 


0-2603 


1-5 


5-0625 


1-5 


1-107 


0-904 


0-667 


0-1975 


1-6 


6-5536 


1-6 


1125 


0-889 


0-625 


1526 


1-7 


8-352 


1-7 


1142 


0-876 


0-588 


0-1197 


1-8 


10-50 


1-8 


1158 


0-863 


0-556 


0-0953 


1-9 


13 03 


1-9 


1174 


0-852 


0-5-26 


0-0767 


20 


16-00 


2-0 


1-189 


0-841 


0-500 


0625 



To find x~'^ the student had better first find x^ and then find its 
reciprocal. The teacher ought to give the student only a few of the 
above exercises on any one night. 

2. Calculate a* and a~^ in the following cases : 







Answers. 




u 








a* 


a-* 


5 


2-43 


49-95 


-02002 


3 


0-246 


1-310 


0-7632 


0-042 


0-476 


0-2211 


4-522 


246-3 


0-2 


3 008 


-3324 


30 01 


2/3 


9-657 


•1036 


002641 


1/7 


•595 


1-681 



14 ELEMENTARY PRACTICAL MATHEMATICS 

3. Calculate the squares, cubes, square roots, cube roots, and reciprocals 
of the following numbers : 



Given 


Answers. 


Numbers. 


Square. 


Cube. 


Square Root. 


Cube Root. 


Reciprocal. 


3931 

227 
8-65 
0-7854 
0-00326 
5-426x10-2 


1545 X 10^ 

51529 
74-82 
0-6169 
1 -063 X 10-5 
2-944x10-3 


6074x10' 
1-170x107 
647-2 
0-4845 
3-465x10-8 
1-597x10-4 


62-70 
15-07 

2-941 

0-8862 

5-71x10-2 

0-2329 


15-78 
6-100 
2-053 
0-92-26 
1-483x10-1 
3-786x10-1 


2-544 X 10-4 
4-405 X 10-3 
01156 
1-273 
306-7 
18-43 



4. Findloge^if logio^=2-1563. 

You are told to multiply the common logarithm by 2-3026, so it is 
evidently good to write 2-1563 in the form -1-8437. The answer is 
- 4-2453. 



5. Convert the following : 



logion 


Answers. 


logion 


Answers. 


log^n 


loge»i 


•2-4822 
4-7995 


-3-4949 
-7-3695 


10213 
3-9812 


-2-2535 
-4-6485 



CHAPTER III. 

THE SLIDE RULE. 

13. To multiply 5623 x 1547 we say • 

log 5623 = 3-7499 
log 1547 = 3-1895 

Adding, we have 6-9394 = logarithm of 8*698 x 10^. 

The essential part of the work is the adding of -7499 and -1895, 
and we can add numbers mechanically in many ways. 

It would be laborious to put 7499 beans in a bag and then put in 
1895 and count the whole. We might put 749-9 ounces and 189-5 
ounces in a scale pan and weigh the lot, thus getting at the sum. 
A good plan is to measure off the distance 7-499 inches or centimetres 
on a scale as in Fig. 1, measure off the distance 1-895 inches or 
centimetres on another scale; and place the two distances so that 
their sum may be measured. 



IC 



Fig. 1.— The distance AB added to the distance BC is the distance AC. 

Now the slide rule does this very thing, only the slide rule has 
not the numbers of Fig. 1 written upon it. It has two sliding 
scales, and in the above position of things the marks upon the scale 
would be those shown in Fig. 2 ; that is, although the distance from 



B 



Fig. 2.— The distances AB, BC, and AC the same as in Fig. 1, but the numbers different. 

A to B represents 7499 to some scale of measurement or other, it 
is not 7499 that is written on the upper scale but the number 



16 ELEMENTARY PRACTICAL MATHEMATICS 



5623, or 5-623 of which ^7499 is the logarithm, and it is not the 
logarithm ^9394 which we read off opposite C on the upper scale, 
although this really is the distance from A to C ; it is the number 
8698 or •8698 which we see there. 

14. Perhaps if I make a slide rule before you it will be easier to 
comprehend. Here are two black scales, A and B (Fig. 3), which 



2 3 4 56789 10 

2-5 3-5 4-5 



!-21-31-4|16'r7181 



fl 



2-5 3-5 4-5 

2 3 4 56789 10 



Fig. 3.— Slide rule manufactured in a few minutes in front of class. 

slide past one another. Making their ends agree, I am going to put 
the same chalk marks on both. I have a sort of tape line here, on 
which I have taken a certain distance (you need not trouble your- 
selves about its actual length in inches) which I shall call a unit 
distance. I have divided it up into 10 and 100 equal parts so as to 
be able to set off any distance less than unity on my scales. To 
know where to put the number 4-5 on my scale I have made its 
distance from the point 1 to be -6532. I have put the number 5 
at a distance from 1 equal to '6990. 



Number on scale. 


At a distance from 
the mark 1 of 


Because 


4-5 
50 

10^0 


•6532 

•6990 

1-0000 


log 4 -5= -6532 
log 5-0= -6990 
log 10=1-0000 



In fact my distances from the mark 1 are proportional to the 
logarithms of the numbers placed at those distances. 

15. A slide rule has several scales. If you have a slide rule 
(Fig. 4), look at the two marked A and B. Slide them so that they 




Fig. 4. 



agree exactly. You will find that the distance from the mark 1 to 
the mark 2 or 3 or 4 or 12 is not 2 inches or 3 inches or 4 inches or 



THE SLIDE RULE 17 

12 inches, but really -3010 inch, -4771 inch, -6021 inch, or 1-0792 
inch, because these four numbers are the logarithms of 2, 3 4, 12. 
So if you place 1 of ^ against 3 of ^, and look for the number on A, 
which is opposite 4 of B, you will see 12. You have multiplied 
3x4 because you have merely added -4771 and '6021 to get 1-0792. 
Think it out for yourself ; practise multiplying simple numbers ; ask 
nobody to help you, and you will rapidly get familiar with and fond 
of the slide rule. 

The lecturer manufactured a slide rule in two minutes, sufficiently 
good to illustrate multiplfcation and division. Every student ought 
to manufacture a slide rule, using two strips of paper. 

To divide 12 by 4. Opposite 12 of A, place 4 of B, and the 
answer is on A opposite 1 of B. Again practise division. 

Multiply 4x3x6x7, and so learn the value of the sliding 
marker (called a cursor). When you have multiplied 4x3, put the 
marker at the answer. You do not want to know this answer. 
Now put 1 of 5 at the marker, and remove the marker to 6 of jB 
and so on. If you practise by yourself, you will need no telling. 
Again, if your answer is beyond the end of scale A, slide B back 
until the mark 10 or 100 occupies the place that 1 occupied; this is 
just as if you lengthened scale A. But instructions are of no use. 
Find all this out for yourself. 

How do you find the reciprocal of a number"? Divide 1 or 10 by 
the number ; it is easy enough to do. Notice that when 1 of ^ is 
opposite 3 of ^, then every number of B is opposite 3 times it of ^. 
Thus one position of B enables us to multiply all the numbers in a 
table by the same number. 

Test for yourself your accuracy in multiplying and dividing. 
Note how curiously the scales are divided, and accustom yourself to 
reading oiF numbers quickly. 

In an actual slide rule (Fig. 4), we have not only the scales A and 
B which slide alongside one another, but another pair C and D 
which are alongside one another. These are prepared in the same 
way as A and B, only that 

if the distance from 1 to 2 on scale ^ or ^ is -3010 inch, 
then „ „ 1 to 2 „ C or D is -6020 „ 

You will see, therefore, that a number on D is just underneath its 
square on A, and you will see that the sliding marker M enables you 
to find squares or square roots. 
P.M. - B 



18 ELEMENTARY PRACTICAL MATHEMATICS 

Also it is easy to multiply or divide any number by the square or 
square root of any other number, and this again ought to be 
practised. So again, any number may be multiplied by its own 
square, and so we get its cube, and the student will see that we can 
reverse this process and so extract a cube root.* Read no book of 
elaborate instructions, you can find out everything for yourself by 
using the rule. Here on the lecture table are numerous contrivances 
called by all sorts of names, which are really all slide rules. Notice 
how the Fuller rule, instead of being straight, is spiral, so that we 
get what is equivalent to an exceedingly long rule in a small 
compass. Also these tables of Professor Everett are really a very 
long rule in the shape of two sheets of cardboard with slits in them. 

16. When students are able to compute a x 6 or a -r 6 or a*, it is 
usual to give them about thirty exercises all of the same kind. 
They have their rule : to apply it needs no thought, and for the 
time being they are machines. Now I do not altogether object to a 
few examples of the same kind coming immediately after one 
another, but I submit that we ought as much as possible to vary the 
kind of question worked in any one lesson. For example, in one 
lesson I would ask a boy to compute 37-56 x 18*23, 37-56 -f 18*23, 
37-56^"^, 37-56"^^, and along with these I would give quite different 
examples. If this is done, a boy has really to think of what he is 
doing all the time. 

It is for this reason that I like to have boys compute from formulae 
quite early, because each part of the work requires somewhat different 
thought from what the other parts require. 

When a boy can multiply and divide accurately and readily he of 
course gets to do these things mechanically, and he ought not to 

I *Some years ago I designed a slide rule to calculate a*. I found that 

f Dr. Roget had used the same method in 1815. My rule is made by Messrs. 

Thornton of Manchester. Here is the principle : 

If a; = a^ log a: = & log a, and log (log a;) = log 6 + log (log a). Keeping scale 
(7 as it is, I substitute for scale D one in which the distance to the mark a 
represents log (log a). If 1 of scale G be placed opposite a of scale i), then 
opposite h on G will be found x on scale D. Even if b is less than 1 it is easy 
to find X. 

The student ought to practise finding such things as 2^, 2^, etc. , or 3^, 3^, etc. , 
of which he knows the answers. 

Evidently we can find the logarithm of any number to any base. We can 
find x = a^l'' or x = a}l'^ at one operation, and there are many other uses of 
this new slide rule. 



THE SLIDE RULE 19 

have to think about the details of such processes ; his thought ought 
to be about the main question on which he is engaged. 

EXERCISES. 

1. Write down the common_ logarithms of I'll 5; 1,115,000; 
0-000,011,15. A71S. 0-0472, 6-0472, 5-0472. 

2. Compute by logarithms and by slide rule (where convenient) : 

68 37x0-002942 ... 54-36x763x273 ^ 

^^ 52-49 ' ^ ^ 760x295 

^...^ 60-8 X 525 xlO^ ,. . 5-306x0-07632 



57750x754x0-7' ' 73-15x^0^02164 

0-1639x52100x0-0253 



(v) \/251 X 



0-00035 X 1 -0264 



, J. 0-0065 X VT36 . , J.. 5-473 x 2-517 x ^3-5 



(0-1324)2 X 0-005621 ' ' ' 1-324 x 4-768 x (l-69)^-'^ 

(^...^ 5-397 X 307 X 760 X (-589)^-^ ^.^^ 97-43-f(0-3524 + 6-321f ^ ; 
^ ^ 273x673x^1-589 • 

(x) (1-342 X 0-01731 -^0-0274)o-3i9. 

Ans. (i) -38320 ; (ii) -2089 ; (iii) 10,472 ; (iv) -03763 ; (v) 9-528 x 10« ; 
(vi) 771 ; (vii) 1-858 ; (viii) 1-448 ; (ix) 0-7557 ; (x) 0-9487. 

3. Evaluate, using logarithms : 



(a) / 31-2X 0-064 ^^. ^^^ ^^^^ ^^^^ ^^ 1782^0-3152. 
^ ^ \ 25-7 X 18-3 ^ ^ 

Ans. (a) 0-06516 ; (b) 17-81 

4. Find the logarithm of 3 to the base e, where e= 2-718. J W5. 1-0986. 

5. Compute, using logarithms : 

(a) 5-36-1-21, 0-2362-' ; (b) 4^237, 10-523 . 

(c) 26-123, 26-12-3; (6^) 0-3^; (e) 3-59"^, -0359*. 

>^ (/) 71-34^8, 0-06321-78 ; (g) S^'^a, 0-3241"-236 ; 

^ ' (h) 25*, 0-0250-3, .25-1-3 . (^) 52-4:5^ 3-0-216^ aiid 0-0420--''6. 

^Tis. (a) 0-1311, -03126; (6)2-985, 223*2; (c) 17820, -00005612; 
(d) -6694 ; (e) -6531, -3299 ; (/) 10-99, 0-007333 ; (g) 17-21, -7665 ; 
(h) 2-924, -3307, 6-063 ; (k) 49-95, -7632, 0-2211. 



CHAPTER IV. 

EVALUATION OF FORMULAE. 

17. Mathematical symbols are merely a very easy form of short- 
hand ; they usually instruct us to perform certain arithmetical 
operations. Thus + , - , x , and -r are well known. Of course, 
axh means the same as a . 6 or a6 when letters are used to represent 
quantities, but only the first of these ought to be used with numbers, 

for very obvious reasons. Again a-rh or a:h ov j^ov a/b mean the 

same. The use of brackets ought to be familiar to students. 
Again J a is the same as a^'; Va the same as a^ ; a^ the same as 
XI a^ ; or by oF^^ we mean the q^^ root of the p^^ power of a or the 
y^ power of the q^^ root of a. Again a~'^ means 1 -r a"*. 

Sometimes in a formula we find a term like sin d or cos 6 or 
tan 6, and this means that, 6 being some given angle, we are asked 
to look up the arithmetical value of the sine of this angle, or its 
cosine, or its tangent, in some suitable table, and use it in calculating 
with the formula. 

Again, sin'^a means " the angle whose sine is a" and if we know 
a it is easy from tables to find sin~ia, cos"^^, tan"^^, etc. It is a 
great pity that this symbol should be liable to be taken for another. 

Again, if we use Xog-^^x we know that we are expected to look 
up the value of the common logarithm of the number re in a table of 
common logarithms or logarithms to the base 10. 

Again, if we see \og^x we know that we are required to use the 
Napierian, or, as some people call it,- the Natural, or, as others very 
absurdly call it, the Hyperbolic Logarithm of the number x; a 
mathematician usually means log^a; when he writes \ogx. To 
convert common into Napierian logarithms multiply by 2*3026. 



EVALUATION OF FORMULA 21 

Some formulae tell us to look up other things, but there are 
always sufficient instructions to enable the necessary arithmetical 
work to be carried out if a teacher will only give exercises to his 
pupils and say nothing of philosophical difficulties which exist only 
in his imagination. The pupils have usually enough common sense 
to follow the plain instructions of even complicated formulae. 

The tables of Chap. VIII. will enable the following exercises to 
be worked. But some of the arithmetical work will be more easily 
done after Chapter V. is read. 

We often use other symbols in our work: a>h means "a is 
greater than h" -, a<b means "a is less than b" ; yccx means "y 
varies as x, or y is proportional to x ; that is, y is equal to x multiplied 
by some constant number." The Greek letter s, which is 2, is often 
used to mean " The sum of all such terms as . . ." 

It ought to be made clear to students that the evaluation of a 
formula is merely arithmetic; there is nothing difficult about it. 
When, by working many exercises, a student has lost his fear of 
formulae and knows that he can evaluate any formula, he will find 
Mathematics an easy stud3^ 

The most essential idea in the method of study called Practical 
Mathematics is that a student should become familiar with things 
before he is asked to reason about them. He must therefore become 
very familiar with algebraic formulae by working such exercises as 
the following. 

Once for all — these four chapters are merely about easy arithmetic. 
When a man uses a complex looking formula, it pleases his vanity 
to think that he is working in complex mathematics. A druggist 
when he makes up a prescription has this same vanity. The man 
who blows the bellows for an organ may have the same. In these 
modern days science has given us the most wonderful machinery, 
and every man who can turn a handle has the same vanity. I 
would impress on students that it is very easy to turn handles, 
and if one is not vain, if one is modest, if one desires knowledge, 
the turning of the handle may lead to a knowledge of the machine 
and to scientific discovery. Indeed, to use another metaphor, the 
turning of the handle is absolutely necessary if we wish the door of 
the scientific mansion to open for us. 



22 ELEMENTARY PRACTICAL MATHEMATICS 



EXERCISES. 

1. If m = {l+\oger) jr, find m for the following values of r : 



r 


1-333 


1-5 


2 


3 


5 


8 


12 


20 


Answers m 


0-965 


0-937 


0-846 


0-70 


0-522 


0-385 


0-290 


0-20 



2. If m = (sr~'^-r~^)/(s-l\ calculate m when s has the values 0-8, 0*9, 
I'l, 1-2 for the following values of »• : 





Answers. 


r 


m 


HI 


m 


m 




when 5=0-8. 


when s=0-9. 


when 8 = 1-1. 


when «=l-2. 


1-333 


0-971 


0-970 


0-962 


0-959 


1-5 


0-948 


0-941 


0-933 


0-926 


2 


0-871 


0-859 


0-834 


0-823 


3 


0-743 


0-721 


0-681 


0-662 


5 


0-580 


0-549 


0-497 


0-475 


8 


0-447 


0-414 


0-360 


0-337 


12 


0-352 


0-318 


0-267 


0-246 


20 


0-257 


0-225 


0-180 


0163 



Notice that you can make no calculations if s is 1. There is a way 
of doing this, however, and the answers obtained are those given in 
Exercise 1. 

3. If u cubic feet is the volume of 1 lb. of dry saturated steam, whose 
pressure is jo lb. per sq. inch, experiment shows that, very nearly. 

Calculate ic for each of the "following values of p. 

For a certain purpose it is found necessary to use the incorrect formula 

Wl = l-^(0•0171+0•0021p). 
Calculate ii^ for the same values of p, and state the error in each case : 



Given values 
oip. 


Answers w. 


Answers Ui. 


Error Mj - u. 


80 


5-36 


5-40 


0-04 


120 


3-66 


3-72 


0-06 


140 


316 


3-21 


005 


180 


2-50 


2-53 


003 


220 


2 07 


2-09 


002 


280 


1-65 


1-65 


0-00 

r 



4. Find p for the following values of «^ using the formula 



Given values of u 


40 


20 


10 


5 


3 2 


Answers p 


9-436 


19-73 


41-28 


86-34 


148-7 


229-1 



EVALUATION OF FORMULA 23 

The following two exercises are on very important approximations : 

5, When a is very small compared with 1 we may write (1 + «)" = 1 + w<x, 
very nearly. Employ this to show that the following relations are nearly 
correct : 

(1-001)3= 1-003; (1-01)^ = 1-0033; (0-99)2 = (1 - -01)2 = 1 - -02= -98 ; 

1 1 



•99 1--01 



=(i--oi)-i=i+-oi=i-oi. 



r^=(l + -01)~^=l --0033 = -9967, 



^^1-01 

\/99 = \/l00(l - -01) = 10(1 - -01)*= 10(1 - •005) = 9-95. 

Find what the exact percentage errors are in these answers. For 
example, find the cube of TOOl by multiplication, and observe how slightly 
it differs from 1-003. 

6. How much error is there in the assumptions 
1+a 



l + i« 



= l + a-/3; (l+a)(l-/?)=l+a-^ 



when 



a=-01, i8=-01; a=--003, /8=--005? 
Ans. No error ; -01 per cent., -001 per cent., -0015 per cent. 



7. Instead of calculating a + 'Ja'^ + h'^, we often substitute, as giving 
nearly the same answer, 1 -84a + 0-846. Take a = l, and, taking various 
values of 6, show the two sets of answers in a table. Within what 
limiting values of h may we assume the error to be less than 3 per cent. ? 

8. P is the horse-power lost by friction when a disc D feet in diameter 
revolves n times per minute in an atmosphere of steam of the absolute 
pressure p lb. per sq. inch. 

Calculate P for the following values of p, D, and n : 



D 


n 


V 


Answers P. 


5 
5 


1000 
1000 


15 

1 


4-7 
0-3 


1 
1 


1000 
1000 


15 
1 


0-0015 
00001 


5 
5 


500 
500 


15 
1 


0-6 
04 



9. A point A is h^ feet above sea-level. A body is projected upwards 
from A with a velocity of Vq feet per second, its height h above sea-level 
t seconds after starting and its velocity upwards are 
h = hQ + VQt-\gt\ 
v==Vo-gt. 



24 elp:mentary practical mathematics 

If /?o = 100 and ^^ = 80 and ^ = 32*2, calculate h and v for the following 
values of t. Here are the answers : 



t 





0-5 


1 


2 


2-484 


3 


4-0 


4-968 


h 

V 


100 

80 


135-98 
63-9 


163-9 

47-8 


195-6 
15-6 


199-4 



195-1 
-16-6 


162-4 

-48-8 


100 

-80 



Besides having the above vertical velocity, the body moves so that its 
horizontal distance x from the point A is ut ; take w = 100 and calculate 
X for the above values of t. Later, plot x and h on squared paper to get 
the real path of the body. 

10. In a manufacturer's list of Thomson turbines, if P is the total 
horse-power of the waterfall, H the height of the fall in feet, n the 
revolutions per minute, R the outer radius of the wheel, 

n = 22-75H^F~'^ and /2 = 2-373P^^"^, 
find n and R for the following values of F and R: 



p 


H 


h 


M 


74 


250 


2629 


0-3'247 


50 


100 


1017 


0-5306 


50 


50 


427-8 


0-8924 


70 


25 


152-0 


1-7758 



Other exercises of the above type will be found at the end of Chapter V. 



CHAPTER V. 
ALGEBRA. 

18. Just as it is quite easy for a beginner to learn to calculate 
from a formula, so it is usually easy for him to do many other useful 
things with formulae. 

Elementary Algebra is made difficult by the mere statement of 
rules. Why should any fuss be made over addition, subtraction, 
and multiplication? Why, anybody who has used a formula with 
brackets knows these things already. 

Tell a boy about ghosts, and the simplest things become complex 
and mysterious. Tell a boy that he is sure to find difficulty in 
simple Algebra, and of course he finds great difficulty with a pro- 
blem that would be quite easy if you told him that it was easy. 
I would give a boy simple equation work at once, and especially 
problems which are solved by simple equations, for there is not 
much that makes a boy think for himself so quickly as such work. 
I know that the average boy will learn quite rapidly and understand 
well, simple equations in «, and also in x and y, and the problems 
leading to them. 

He ought to be able to state in words the meaning of an algebraic 
formula if he has done such exercises as I have given in Chapter IV., 
and he ought to write out algebraically any rule that is stated to 
him in words. Let him practise this a little, and he will presently 
laugh at the difficulties which teachers find in making their students 
work problems. At the same time, simple problems will give him 
an interest in the work. I say, problems leading to easy equations. 
Complex-looking equations seldom arise in practical work, the 
exercises leading to them have been created by our examination 
systems. They are generally tricky, and are meant to test if a boy 
can work very rapidly without making mistakes, but really for the 



26 ELEMENTARY PRACTICAL MATHEMATICS 

working of a practical question a man has plenty of time. Easy 
exercises like the following ought to be set : 

Ex. 1. Divide 8-37 into two parts, one of which is 2-4 times the 
other. Here the two parts may be taken to be x and 8*37 - x. Let 
us say then that 

8-37-a; = 2-4a; or 3-4a; = 8-37 or a;=2-4618 
and 8-37 -2-4618 = 5-9082. 

Ex. 2. Find two numbers whose sum is 22*5, and whose 
difference is 5-6. Let x be one of them and 22-5 -a; the other. 
Then 22'5-x-x = 6'6 or a; = 8*45. 

Ex. 3. A father is 3-5 times as old as his son; in 20 years he 
will only be twice as old. What are their ages now 1 

Let X be the son's age, the father's is S'bx. In 20 years their 
ages will be 3-5a; + 20 and a; + 20. The question states that 
3-5a; + 20 = 2(a; + 20). 

From which we find ic= 13-33 years; this is the son's age. The 
father's age is 3-5 x 13-33 or 46-67 years. 

Ex. 4. Divide a number 8*32 into two parts such that 5 times 
one part, subtracted from 6 times the other, leaves 1-56. Let x be 
one part, 8*32 - a; is the other. Then 

6a; -5(8-32 -a;) = 1-56, 
6a; -41 -6 + 5a; =1-56, 
lla; = 43-16, 
x= 3-924. 
The parts are 3-924 and 4-396. 

Sometimes a problem may be solved more easily if we use 
simultaneous equations. Take Ex. 2 above. Let the numbers be 
X and y, and the question tells us that 

x + y = 22'5, 
x-y= 5-6. 
Adding these equations, we get 2a;, or, subtracting them, we get 2y. 
Many problems lead to quadratic equations ; that is, equations of 

the form x'^+px + q = (1) 

The rule for working this is ; — write the equation as 

x^+px= -q. 
To both sides add the square of half ^ or ^, 

x^+px + ^=^-q. 
This makes the left-hand side a perfect square. 



ALGEBRA 27 

Extract the square root of each side; and because when we 
extract the square root of, say 9, the answer is either +3 or - 3, 
we write it ± 3. Hence 

P 



^ + 2 



p ip-i 



(2) 



There are, then, two answers, or roots as they are called, to a 
quadratic equation. 

If we split the expression x^-{-px + q in (1) into factors, let us call 
them x-a and x-b, so that (1) is 

{x-a){x-b) = 0. 

Now, how can this expression be 1 It is ii x - a = or ii x = a ; 
and it is ii x-b = or x = b. 

If we see an expression like ic^ + 1 1^^ _|_ 30^ and if we are a little 
familiar with formulae, we know that its factors are x + 6 and 
x + 5, because our eyes get accustomed to the idea that 30 is the 
product of 6 and 5, and 11 is the sum of 6 and 5. In fact 
{x -a){x-b) = x^ -{a-\-b)x->t ab. 

Or if {x - a){x - b) is to be the same as x^ +px + q, then q is ab 
and -p is a + b. 

In many cases mere inspection does not help us, but the above 
reasoning tells us what to do. We have to find the factors of 

x'^ +px + q. 

Put it equal to and find the roots as in (2). If these roots are 
a and b, then x-a and x-b are the factors. 

Example. Find the factors of «2_ s-g/g- n-g. Put it equal to 
and find the roots, either by the method which led us to rule (2) 
or using (2) as a formula. In this case p= - 3*8, q= - 11-6. The 
roots are then 1-9 ±>yi-92+ 11-6 = 1-9 ±3*9 or 5*8 and -2. The 
expression is then the same as 

(x-5'S)(x+2). 
In solving quadratics we sometimes find in our answer such an 
expression as V - 25, which is called an impossible or an unreal 
quantity. Although we cannot at present attach a meaning to such 
a thing, it is good algebra to work such questions, giving the unreal 
answers. To save trouble we write V- 25 as n/25 xJ -I ; we often 
use the letter i for J~^, so that J -25 is 6i. 



28 ELEMENTARY PRACTICAL MATHEMATICS 

Solve the quadratic x^-4:X + 5*44 = 0. 

We find x = 2±l"2i &s the two roots. 

In Chapters XXVIII. to XXXVI. the student will find that we 
can make an important practical use of unreal quantities. 

When the student gets an equation to solve, say f{x) = 0, he will 
now know that if, by guessing or by the use of squared paper or in 
any other way, he has found the answer, say ic = a, then x-a must be 
a factor oif(x). If now he divides /(a:) hy x - a, he may get a simpler 
equation to solve ; if he finds x = b, then x-b must be a factor of 
f{x). If he again divides hy x-b, he may get a still simpler equation, 
and so on. 

Example. Solve the equation x^ + 5-lx^+2'dx-9 = 0. Let us 
suppose that by guessing, or by the use of squared paper or in some 
other way, the student finds that x=l is a root of the equation. 
Now let him divide by x-l, and he finds x"^ + 6-lx+9 = 0. The 
roots of this quadratic are x= - 3*6 and x= -2-5. Therefore the 
three roots of the given equation are 1, - 3*6, and - 2*5. 

A little, and only a very little, work on simultaneous quadratics 
ought to be given. 

Example. x^ + y^ = 3l-lS, x + y = 8-4:0. 

Squaring the second equation and subtracting the first, we get 
2xy = 33*43. Subtracting this from the first equation and extracting 
the square root, we get x-y= ±1-92. Adding this to the second 
equation and dividing by 2, we get a: = 5"16, y = 3'24:, and also 
x=3-24:, «/ = 5-16. Hence if the question had been: — Find two 
numbers such that their sum is 8*40 and the sum of their 
squares 37*13, our answer is 5*16 and 3*24. 

Example. If x'^ + y^ is given and also xy, we 4iave only to add 
and subtract 2xyj and in each case to extract the square to get 
x + y and x-y. 

Example. x^-y^ = S7 and x-y = l. Divide, and we have 
x'^+xy + y^ = 37, also x^ - 2xy + y^=l. Subtracting, Sxy = 36 or 
xy=12. }lencex^ + 2xy-{-y^ = 4:9 or x + y= ±7. Hence « = 4, 2/ = 3 
or a? = - 3, ^ = - 4. 

In the course of the above work, and not as a separate kind of 
work, the student will simplify expressions; he will intuitively 
recognise the factors of such expressions as 

^2-144, x'^-6x-66, or a;2 + 7a;+12. 

He will multiply and divide without making any fuss about the 
matter, as if he were dealing with new and independent subjects. 



ALGEBRA 29 

In such multiplication and division he ought to try to remember 
the answers to {a + xy when n is 3, 4, and 5 ; 

also the series which are answers to r. and ^ . 

\ -X 1 +x 

Where the greatest error is made in teaching is in not intro- 
ducing to a student, quite early in all this work, as a sort of 
relief work, the plotting of functions by means of squared paper. 
For if he takes any function of x and calls it y, and for any value 
of X calculates y and plots the corresponding values of x and y on the 
squared paper, he gets a curve. If he now notes the value of x 
which makes the function 0, he gets in the simplest fashion the 
very best knowledge of the solution of an equation, of the root, of 
the roots of an equation, and he can and will philosophise on the 
subject without any prompting from his teacher. 

I must confess, however, that the compilers of modern school 
algebras must make the gods laugh over the uses to which they put 
this plotting of functions. 

Just here it is well to point out that when we have a formula of 
several algebraic quantities, it is often easy to express any one of 
the quantities in terms of the others so as to make many useful 
calculations. In fact, then, I would ask teachers to mix together as 
one simple kind of algebraic work, easy to give, even to beginners, 
many parts of Algebra which are usually taken up very much later, so 
much later that the average student never reaches them in his study. 

It will be seen, then, that I include in such work all sorts of 
work which goes under the name of " Rules in Arithmetic." 

The following are only a few of the many hundreds of examples 
that may well be put before students : 

EXERCISES. 

1. If Q^O'OOMbdHn, find d, having given ^ = 4800, l = % m = 125. 
Ans. 59-36. 

W hd^ 

2. If y=iol^jZMEI, where z^? = -^ and I=-j^, find u\ /, and y if 

IF=.3-5x2240, ^ = 147, 6 = 3, (^=9, ^=1-1x106 

Ans. w=53'33, /= 182-25, y =0-3234. 

3. The energy stored in similar flywheels is E=ad^n\ when d is the 
diameter in feet, n the speed in revolutions per minute, and a is a 
constant. If when d=b and ?i = 100, E is 18,500, find a for that kind 
of flywheel. Find d for a similar flywheel if E increases by 10,000 
when n increases from 149 to 151. Ans. a = 0000592, 6^=7-761. ^ 



30 ELEMENTARY PRACTICAL MATHEMATICS 

4. If x'lf^ — a ; if ^ is 5 when y is 10 and also if ^ is 11 when y is 8, 
find n and a. What is the value of y when a: is 7 ? 

Am. ?i = 3-533, a = 17060, 9-09. 

5. In a miner's handbook the following formula is given for the 
thickness of a cylindrical dam : 

^ = r(l-\/l-20p/f). 
If r=144, 7^=1-075 x 10^ and 144p = 20 x 62-5, find t. Am. 0-116. 

6. If ^/y = e«*, where 6 = 2*718; if a = 0'3 and ^=2-85, and if .r- 3/ = 550, 
find X. Ans. 957-0. 

7. When a; and y are small, we may take (1 +^)/(l +y) as being very 
nearly equal to \-\-x-y. What is the error in this when ^•=0-02 and 
y=0-03 ? Am. 0-000291 or 0-0291 per cent. 

8. y = ax^-\-hofi ; when .r is 1, y is 4-3, and when x is 2, y is 30. Find 
a and h. What is y when ^ is 1-5 ? Aris. y = \'\x'^-\-Z''±3p'^ 13-275. 

^ ^. 1 853 . ^ ^ 836 , 677 , 961 

9. If iog,^ + 0-9x — = log.^ + g;^.r, 

find the value of x to three significant figures. Ans. 0*784. 

f' 10. Use the formula tan — ^ rtan — ^r — = r to find the angles 

/ 2 2 a-h ^ 

f A and B of the triangle ABC, given that C=29°-7, a = 86-92, 6 = 54-68. 

Am. 115°-5, 34° -8. 

11. If 9 = j{Y^ + ^ogJj)> if ^^ = ^ + 461 and t, = 0, + 4ei', if l=ff-h 

and l^ = H^-h^; if ^ = 230 and ^^ = 338 ; if jy=1152-l and A = 198*7; 
if H^ = 1185*0 and A^ = 308-7, and if q^ = 1, find the value of q. Ans. 0*9. 

12. In a thick cylinder subjected to fluid pressure the radial com- 
pressive stress jo and the hoop tensile stress / at a point which is at the 
distance r from the axis are such that 

V = ^.-^^ (1) 

/=^ + «, (2) 

where a and h are constants. 

(a) The internal pressure 'p^ is 3 tons per sq. in., the inside radius r^ 
being 5 inches ; the outside pressure po is 0, the outside radius ^q being 
10 inches. Find a and h in the formulae (1) and (2), and calculate p and/ 
forr = 5, 6, 7, 8, 9, and 10. 

Here 3 = A-a, 

Subtracting, we find 3 = *036 or ft = 100 ; therefore a = \. 



L 



(1) and (2) become p = ^-l, f=-^ + l. 



ALGEBRA 



31 



r 


P 


/ 




, 




5 


3 


5 


6 


1-778 


3-778 


7 


1041 


3 04] 


8 


0-5625 


2-56-25 


9 


0-2346 


2-2346 


10 





2 



Later, the curves for jo and /ought to be plotted on squared paper. 

Instead of taking numbers for p^ and r^ and 7'q, let the student now 
repeat the above work using these letters, giving p and / in formulae 
applicable to all such cases. TakejOo=0. 

((3) A gun tube ?o=15, ri = 13 is to have po=0,/i=20 ; findpj and/o- 
= -|- _ a = 0-0044446 - a, 



20=— - + a = 0-0059176 + a. 
Iby 

Adding, we get 20 = 0-0103616 or 6 = 1930-3, a = 8-5783, 

1930-6 



Pi = 



169 



8-58 = 2-84, 



The student ought to give general formulae for this case also. 

(y) Agun tubero = 13, ri = ll istohavepo=2-84,/i = 20; find jOj and/o. 

Ans. pi = 6-e2, /o= 16-22. 

(5) A cylinder, whose 7\ is 5 inches and r^ is 10 inches, has an inside 

/i = 10000 lb. per sq. inch. If p^ is 0, what is pi ? What is / everywhere ? 

what is jt? every where ? ^ rv , ^ j tr^r^r^r^ . ^ 

^ ^ Ans. 0=-a-{-ir~ and 10000 = a + xr. 

100 ^O 

We find from these equations a = 2000, 6 = 200000, and hence 

200000 
/=2000H 2 — everywhere. 



The inside p is 6000 lb. per sq. inch, 



everywhere. 



Show / and r and also p and r in two curves. 

19. I want a student to practise using all sorts of formulae, so 
that he shall cease to be afraid when he sees one. Of course, there 
may be some bit of shorthand, some symbol, which has not been 
explained yet to him, but he ought to know that there is nothing 
magical or uncanny about it. I have taken up some formulae at 



32 ELEMENTARY PRACTICAL MATHEMATICS 

random. I might call any one of them a rule, and so create 
difficulty, but indeed there is only one way with them all. 

The average man who has worked through many rules in complex 
arithmetic, and algebra, and engineering, very quickly forgets them 
all, except the one or two that he constantly needs. , It is only a 
teacher who remembers hundreds of rules. But if at the beginning 
a man knows that his rules are all one rule; all his separate rules 
are mere examples of one general principle ; he never can forget it, 
for every common-sense calculation that he makes only fixes the 
general principle more firmly in his mind. 

Have you not noticed that a great man has only a few simple 
principles on which to regulate all his actions'? A great engineer 
keeps in his head just a few simple methods of calculation. But 
note that, through constant practice, the simple principles or 
methods are always ready for use in his mind. It may be that 
an expert may be quicker or neater in working some one kind of 
problem, but however clumsy or tedious may be the great man's 
method of working the problem, he gets the right answer, and he 
has no misgiving as he writes it out. 

My one simple rule is to treat all numerical calculation as work 
with some formula, and all rules ought to be compactly stated as 
formulae. 

Of course, it is another thing to work out such formulae, to prove 
them correct, and yet I say that, even here, my general principle 
introduces enormous ease, for, in most cases, to understand and feel 
unafraid of a formula is almost to see the proof of it. 

20. Notice that if we have any equation, say M=N, we can say 
that \ogM=\ogN or M'' = N'', where n is any number. Again, if 
x^ = 'ip we may say that n log x = m log y. 

Ex. 1. If ^u^"^ = 479, when u = 4, find p. 

Here \ogp + 1 -0646 log 4 = log 479, and we find p = 109-5. 

Ex. 2. Again, when ^ = 203, find u. Ans. 2-24. 

Ex. 3. Suppose ^v^'^^ = ft. 

(a) If ^ = 100 and v=l, find a. Ann. 100. 

If V becomes 1'5, using the same value of «, find^. Ans. 63*24. 

(P) If V becomes 2, 2J, 3, 3J, 4, in each case find the correspond- 
ing value of p. 

Repeat the above work w\\en pv^'^ = a, taking ^ = 100, and v= 1 to 
start with. 



ALGEBRA 



33 



(y) Repeat again when pv = a. 

Show the three sets of answers for ^ in a table. 

Ex. 4, lip^v^"^'^=p^v^'^^ and if v^jv^ be called r ; if ^2 = ^* fi"^ '' 
for the following values of j9j : 



Pi 


250 


200 


150 


100 


50 


Answers, r 


27 13 


22-27 


17-26 1206 


6-53 



21. Proportion. The Ratio of a to h or a:h is a -r i. Thus the 
ratio of 2 to 3 is 0-6667. 

When we say that y varies as, or is proportional to x, we write 
the statement in the form y cc x, and we know that this is the same 
statement as y = ax, where a is some constant number. 

Simple Propoi'tion. li y oc x and if y = 4 when x = 3, then as y = ax, 
4 = a X 3, or a = f , and hence y = ^x is the true law connecting 
y and x. If any value of x be now given, y may be found. 

Compound Propaiiion. If 10 labourers dig 150 yards of trench 
in 5 days of 12 hours; how many labourers will dig 356 yards in 
2 days of 9 hours] If we use /, y, d, and h, we see that the 
assumption made is 



dh 



dh' 



Hence 



10 = a 



150 ,, , 5x12x10 , 
-, so that a = r^^ = 4. 



5x12' 



So that we have the formula / 
Thus the answer wanted is 



150 
4-^ for working any exercise. 



356 
Z = 4^„=^ 79-11 labourers. 
J X y 

22. Now notice that all the following exercises are worked in the 
same way as those of Art. 21. 

Ex. 1. In steam vessels of the same character, / being indicated 

horse-power, D displacement, v speed ; / <x D^ v^ \ 

(a) If 7=655 when i> = 1720 tons and v=\0 knots, find the 

exact rule. Ans. /= i)^ v^ ^ 219. 

iP) If D is 1500, V = 13, find /. Ans. 1314. 

(y) If 7=800, 7>= 1300, find v. Ans. 11-37 knots. 

At the highest speeds of modern ships, where the wave-making 
resistance is much more important than mere skin or eddy-making 
resistance, we may take Fronde's law to be true for the whole 
P.M. C 



34 ELEMENTARY PRACTICAL MATHEMATICS 

resistance, and say t hat, at corresponding speeds (t hat is, speeds 
proportional to the sg[uare roots of lengths of ships), the indicated 

power being / and displacement D, I=c D^, where c is a constant. 
Thus if a cross- Atlantic liner of 20 knots, i) = 10000, 7=20000, 
crosses in 6 days ; 

If we want to cross in 5 days we must have «? = 24 knots, and as 

V oc x/length oc Z)\ D must be increased to 10000 x (|)6 = 29860 
tons and I to 20000 (|)7 = 71660. 

The total coal used in passage oc D, that is, it is always the same 
fraction of the displacement, 

Ex. 2. The weights PF of similar objects are as the cubes of 
their like dimensions and as the densities of their materials. Two 
similar objects are in the linear proportions of 1 to 4*37, and their 
densities are as 2 to I "74. The weight of the first is 20 lbs. What 
is the weight of the second 1 Ans. JV= 1452 lbs. 

Ex. 3. If d is the calibre of a gun, it is usually assumed that 
the weight is proportional to d^, and that the thickness of armour 
which its projectile will pierce is proportional to d. If an 8-inch 
gun weighs 14 tons, and can pierce 11 inches of armour, what 
thickness will be pierced by a 10-inch gun, and what is the weight 
of the gun? Ans. 13-75 inches; 27*34 tons. 

Ex. 4. Two models of terrestrial globes are 1*22 feet and 3*14 
feet diameter respectively. If the area of a country is 15 square 
inches on one, what is it on the other? Ans. 99-32 square inches. 

Ex. 5. If ^x h^ and if ^ = 0-466 when A = 0-5, find the true 

law. Ans. ^ = 2-636^1 

(a) Find Q when h=\'X Ans. Q = 5-079. 
ifS) Find h when Q = 2-46. Ans. h = 0-9727. 

Ex. 6. If we know that y = a + bx, where a and b are constant 
numbers : 

(a) If ^= 12 when x=l, and if «/= 15 when x=^6, find a and b. 
Here we have l2 = a-\-lb, \5 = a + 5b. 

Subtracting, we have 

3 = 4^>or^> = |, 12 = ft-f-| or a= llj. 
Hence the law is really ?/= llj -ffic. 
(^) Find?/ if a: = 4. Ans. 14^ 
(y) Find x when 2^ = 20. Ans. 11|. 

Ex. 7. liy = ax + bzh^, 

if ^ = 49-5 when a; = 1 and ^ = 8, 
and y = 356 when a: = 1 -5 and 2; = 20, 
find a and b, and find the value of y when « is 2 and z is 17. 

Ans. y= -57-la; + 26-65A2; 590*7. 



ALGEBRA 35 

23. Those parts of arithmetic called "Equation of Payments," 
" Barter," " Profit and Loss," " Fellowship," " Alligation " of many 
kinds, "Position," "Double Position," "Conjoined Proportion," and 
many others, are, when we strip them of their technical terms 
and artificial complexities, the simplest of algebraic exercises, and 
they ought to be treated as such. 

24. Aritlimetical Progression. It is easy to show that 

l^f+(n-l)d and s = |7i(/4-/),* 
if there are n terms with common difference d (that is, any term 
minus the preceding term), if their sum is s and if / is the first and I 
the last term. 

It is, therefore, very easy to find any two of these when the 
other three are given. 

(1) Find the 15*** term and the sum of 15 terms of 0*25, 0-50, 
0-75, etc. Jns. 375, .30. 

(2) What is the first term when the 59*^ term is 70 and the 66*** 
term is 84 'i Ans. - 46. 

(3) Insert 4 arithmetic means between 3 and 18. 

Ans. 6, 9, 12, 15. 

25. Geometrical Progression. If /, /, s, n, are as in the last case ; 
if r is the common ratio (that is, any term divided by the preceding 
term), it is easy to prove that 

(1) Find the 10*^ term and the sum of 10 terms of 2, 6, 18, etc. 

Ans. 39366, 59048. 

(2) Insert 3 geometric means between 3 and 243. Ans. 9, 27, 81. 

* The student who writes the series down algebraically,/, /+ d, f+ 2d, /+ Sd, 
etc. , sees that the 2°^ term has d, the S'^ terra has 2d, the 4*^ term has 3d, and 
so the 40*^^ term would have 39d and the n^^ term would have {n-l)d. 

Again the sum of the first and last terms is the same as the sum of the 2^^ 
and the 2"*^ to last, of the 3"^ and the S"^ to last, and so it is easy to write 
the sum of all the terms in the shape above given. 

t The geometric series is evidently 

/j /»*> ff*} /^ .../r"~^ if there are n terms. 

Now, by mere division, it is easy to show that 

1 — r" 

=: 1 + ^ + ,'2 + etc. + r^'S 

1 -r 

and the sum of the series is evidently this multiplied by/. 



36 ELEMENTARY PRACTICAL MATHEMATICS 

(3) The 6*^ term of a geometric series is 20*34, r is 0*26 ; find 
the first term and the sum of the 6 terms. 

Ans. 1-712x104, 2-313x104. 

(4) The sum of a geometric series of 5 terms is 534, r is 1*65; 
find the first and last terms. Ans. f= 30-9, I = 229'1. 

26. Compound Interest. It is very easy to prove that if a sum 
of money F (the principal) is lent at compound interest at r per 
cent, per annum, it amounts in n years to 

and the increase I = A - P may be called the interest* 

Ex. 1. If P= 255-75, r = 3|, n=l7, find A. Ans. 459. 

Ex. 2. If ^ = 930, 7- = 4J, n= lOJ, find P. Ans. 585-82. 

Ex. 3. If P = 320, 7=456, r = 5, find n. Ans. 18-156. 

Ex. 4. If P=250, ^ = 420, ?i=16, find r. Ans. 3-3. 

Ex. 5. P = one farthing, n = 1 900, ?- = 5 ; find A. 

Ans. 2 X 10^^ pounds nearly. 

Ex. 6. Find ?i if ^ = 2P. Ans. w = log 2 -f log (\ + ^Y 

Later we shall find by squared paper that % = 70-^r is a good 
approximation, much used by practical men. 

Ex. 7. If interest is added on m times a year, prove that 

l) 



A=P 1 + 



100m, 

Ex. 8. If interest is added every moment, A = Pe*"-/!"". 

Ex. 9. The population of England and Wales doubles itself 
every 50 years ; what is the rate per cent, per annum of increase ? 

A = 2P, n = 50, and hence r = 1 -396. 

li A = 2P we see that nr=69'S. 

27. Present Value and Discount. Use the formula of Art. 26 if 
the sum A is due at the end of 7i years and P is its present value. 
Call A -P the discount. 

* Thus, at 5 per cent, per annum, the interest on a sum of money P for one 
year is '05P, so that if this interest is added to P the amount is 1 -05P. In fact 
a sum of money P at the beginning of a year becomes 1 'OSP at the end of the 
year. Therefore, at the end of the second year, the amount is 1 '05^P, and at 
the end of the third year it is 1 '05^P, and at the end of the n^^ year it is 1 OS^P. 



ALGEBRA 



37 



It is easy to prove that an annuity of a per annum, payable for 
n years, has a present value P, or will amount to A if the use of 
money is worth r per cent, per annum, if 

rA /, r \" , ., rP 



100a 



= 0^4) 



- I, or if 



lOOrt 



1- 



(^^4)'"- 



One of these formulae is enough if it is remembered that the 
connection between A and F is what was given in the rule for 
compound interest. 

28. General Exercises on Formulae. Students are asked to take 
up numerical exercises on all these. Such exercises are to be found 
in many books. But indeed almost any pocket book formula serves 
for the creation of good exercises. 

Above all, I would impress upon teachers that they should 
introduce no artificial difficulties, no tricks, no conundrums into 
algebraic work. Let every exercise be straightforward and as easy 
as possible. 



EXERCISES. 

1. Slipping of Belt on Pulley. If N is the tension on the tight side, 
M on the slack side, 6 radians the angle of lapping, /x the coefficient of 
friction between belt and pulley, then NjM=€f^^. 

(1) If /x = 0'3, find NjM for the following values of 0. What is 6 in 
degrees if 180 degrees are 3"14159 radians? 

(2) FindalsoiV/(/V-3/).* 



e radians. 


e degrees. 


N/M. 


JV/(A'-.V). 


1 


57-30 


1-35 


3-86 


2 


114-59 


1-823 


2-21 


3 


171-88 


2-46 


1-69 


4 


229-18 


3-320 


1-43 


5 


286-48 


4-482 


1-29 


6 


343-78 


6-050 


1-20 


7 


401-07 


8-167 


114 


8 


458-37 


1103 


1-10 


9 


515-66 


14-88 


107 


10 


572-96 


20-09 


105 



jV 
* By Algebra we find that if -ji^=a 

N 



N 



1-35 

N-M~0'3o 



M 



N-M a-\ 



Thus, 



= 3-86. 



if 1=1-35. 



38 ELEMENTARY PRACTICAL MATHEMATICS 



2. Compound Interest. We have the formula 



A 



-( 



1 + 



100 



The money F is lent at r per cent, per annum, and in 7i years A is its 
amount. The interest / is ^ - P. 

(1) Calculate A/F, which is ( 1 +t7T7; ) when r = 5 or 4 or 3 or 2 or 1 for 
the following values of n : 



n 


Answers. 




r=5. 


r = 4. 


r=3. 


r=2. 


r = l. 


10 


1-629 


1-48 


1-344 


1-219 


1-105 


50 


11-47 


7 107 


4-384 


2-692 


1-645 


100 


1-315x102 


5-050x10 


19-22 


7-245 


2-705 


200 


1-729x10^ 


2-551 x W 


3-694x10^ 


5-249x10 


7-316 


400 


2-99 xl08 


6-506 xlO« 


1-364x10-5 


2-755 X 103 


5-353x10 


1000 


1-546x1021 


1-08 xlO" 


6-87 X1012 


3-983 X 108 


2-096 X 10* 


1500 


6-08 xlO^i 


3-543x1025 


1-80 xlO^y 


7-949x1012 


3 035x106 


2000 


2-39 X1042 


1-166x10^ 


4-725x1025 


1-585x101' 


4-393 xlO« 



(2) If yl/P=2, find 71 for the following values of r. This is the number 
of years in which a sum of money will double itself. 



Values of r 


1 


2 


2-5 


3 


3-5 


4 


4-5 


5 


Answers, n 


69-66 


35-0 


28-07 


23-45 


20-15 1 17-67 15-75 1421 



The student will have seen that ?i = 0-3010-=-log( 1 + ^^ j. 

(3) Let us take it that a sum of money will double itself in 1 4 years. 
Suppose that a glass of port was worth sixpence in the year 1800. At 
that time it was placed in a cellar and was not drunk till 1912 ; what is 
now its value? 112 years have lapsed, or eight times 14 years. So that 
the sixpence must be doubled eight times. Ans. 128 shillings. 

3. Exercises, using the Tables, Chap. VIII. 

(1) sin (J + ^) = sin J cos ^ + cos ^4 sin ^, 
sin (A — B) = sin A cos B — cos A sin B, 
cos {A+B)=cosA cos B — sin A sin J5, 
cos {A-B) = cos ^ cos 5 + sin A sin B. 

Using yl=55° and 5=32°, test these formulae. 

„ ^ = 50° „ B = \r 
Write out what the formulae become if A is 90°. 

5 is 90°. 

)> » » ^— ^' 

(2) sinJcos5=|{sin(^+5) + sin(^-5)}, 
sin^ sin5 = |{cos(J - B) - co^{A -\- B)} , 
cosJcos5 = i{cos(^+jB) + cos(^-5)}. 



ALGEBRA 



39 



Using ^4 = 52°, ^ = 15*, test these formulae. 

.4 = 28°, 5 = 12° 
Write out what the formulse become if ^ = 90°. 

^=90°. 

« « It A=jo. 



(3) siii^ + sin<^=2sin^(^ + <^)cos|(^-<^), 

sin 6 — sin cf> = 2 cos ^{d + 4>) sin ^{d — cf)), 
cos ^+cos^ = 2cos^(^ + </))c6s^(^ — <^)j 
cos ^ — cos<^=2sin^(^ + <^)sin^(^ — 6). 

Illustrate these rules when ^ = 67° and ^ = 50°; also when ^=60°, 
(^ = 30° Also when 6 = 30°, <^ = 60°. 

4. Exercises on y=e**; find its value for the following values of x. 
The student already knows that e is 2-71828, the base of the Napierian 
system of logarithms. In this case bx is log^y. 









Answers. Values of e**. 






Given Values 














of a:. 


When 


When 


When 


When 


When 


When 




bis 4. 


6i8l. 


6isi. 


6 is -i. 


6 is -1. 


6 is -4. 


-11 


01227 


0-3329 


0-7596 


1-317 


3 004 


81-45 


-10 


0-01831 


0-3679 


0-7787 


1-284 


2-718 


54-61 


-0-9 


02732 


0-4065 


0-7985 


1-252 


2-460 


36-60 


-0-8 


0-04076 


0-4493 


0-8187 


19^, 


2-2-25 


24-53 


-0-7 


0-06081 


0-4965 


0-8395 


1-191 


2 014 


16-44 


-0-6 


0-09071 


0-5488 


0-8607 


1-162 


1-823 


11-03 


-0-5 


01353 


0-6065 


0-8824 


1-133 


1-649 


7-389 


-0-4 


0-2018 


0-6703 


0-9047 


1-105 


1-491 


4-953 


-0-3 


0-3011 


0-7408 


0-9277 


1-077 


1-350 


3-321 


-0-2 


0-4492 


0-8188 


0-9512 


1-052 


1-221 


2-225 


-0-1 


0-6704 


0-9047 


0-9753 


1-025 


1105 


1-491 


•0 


1000 


1000 


1-000 


1-000 


1-000 


1-000 


01 


1-491 


1-105 


1-0-25 


0-9753 


0-9047 


6704 


0-2 


2-2-25 


1-2-21 


1-052 


0-9512 


0-8188 


0-4492 


0-3 


3-321 


1-350 


1-077 


0-9277 


0-7408 


0-3011 


0-4 


4-953 


1-491 


1-105 


0-9047 


0-6703 


0-2018 


0-5 


7-389 


1-649 


1-133 


0-88-24 


0-6065 


01353 


0-6 


1103 


1-823 


1162 


0-8607 


0-5488 


09071 


0-7 


16-44 


2-014 


1-191 


0-8395 


0-4965 


0-06081 


0-8 


24-53 


2-225 


1-222 


0-8187 


0-4493 


0-04076 


0-9 


36-60 


2-460 


1-252 


0-7985 


0-4065 


0-02732 


10 


54-61 


2-718 


1-284 


0-7787 


0-3679 


0-01831 


11 


81-45 


3-004 


1-317 


0-7596 


0-3329 


0-01227 



Only a few of these exercises ought to be given to any one student. 

5. The Binomial Theorem is easy to prove. It is a pity that students 
are usually subjected to an unnecessary course in Permutations and 
Combinations before this proof is given to them. It is on account of this 
that many artificial and absurd exercises arc given, such as " What is the 
r*^ term ? " or " Show that the r*^ term from the beginning is equal to 
the r*'' term from the end." 



40 ELEMENTARY PRACTICAL MATHEMATICS 

The theorem is :— For all values of x^ a, and n, 
(x + af 



» 1 . n(n — l) 2 « 2 ■ n(n-l)(n-'2.) , ,, 3 



in-l){nr2)(n-3) 

5 "^"^ 



4- etc. 



Here | 4 means 1x2x3x4. Eecently the symbol 4 ! has been used 
instead of [4. 

(1) Write out the theorem for the cases ?i = 2, ?? = 3, ?i=4. 

A71S. {.v + ay=,v^ + 2a.v + a\ 
„ {x + ay=x^ + 3ax^+Za^x + a^ 
„ (^ + a)4 = ^-4 + 4a^ + 6aV'^ + 4a% + a'*. 

(2) Show that (l-{-a)-^ = l-a + a^-a^+a*-etc., 
and that (1 -rt)-i = l +« + a2 + aHa* + etc., 
and test these by dividing 1 by 1 +a or by 1 -a. 

(3) Calculate (l+a^ for the following cases : — Where n = 2,'S ; - 1, -2, 
-3; i, i, j; -i, -J; letting « = -01, -001, --01, --001. 

Only four significant figures are needed in the answers. 
Let 1 + a be called y. 





Answers, 


?f 


v/2 


,,s 


y-l 


?/-2 


y-^ 


?/* 


?/^ 


^^ 


.- 


,,-i 


101 
1-001 
0-99 
0-999 


1-020 
1-002 
0-9801 
0-998 


1-030 
1-003 
0-9703 
0-997 


0-9902 
0-9991 
1010 
1-001 


0-9804 
0-9782 
1-020 
1-002 


0-9709 
0-9983 
1 -031 
1-003 


1-005 
1-001 
0-9950 
0-9995 


1-003 1-003 
1-000 1-0005 
0-9967 i 0-9975 
0-9997 0-9998 


0-9955 
0-9991 
1-005 
1-001 


0-9967 
0-9997 
1-00335 
1-0003 



(4) Calculate (1+a)" in the following cases, giving more than four 
significant figures in the answer. 



Values of y. 


y2 


y' 


^^y 


^ 


1 
y 


1-001 


1 -002001 


1 -003003 


1-0005 


1-0003 


0-9991 


101 


10201 


1 -0303 


1-005 


1-003 


0-9902 


1-1 


1-21 


1-331 


1-049 


1032 


0-909 


1-5 


2-25 


3-375 


1-225 


1145 


0-6667 


20 


4 


8-0 


1-415 


1-26 


0-50 

- 



The values found in this laborious way ought to be tested by ordinary 
arithmetic or by using logarithms. 



ALGEBRA 



41 



6. If e*= 1 +^' + 7-^ + 7— + etc., calculate e* for the following values of x : 



Values of x. 


N or e*. 


001 


1-00100 


01 


1-01005 


01 


1-1052 


0-2 


1-2214 


0-4 


1-4918 


0-7 


2-014 


10 


2-71828 


1-5 


4-4817 


2 


7-389 



X is here the Napierian logarithm of N, and when ^ is 1 we find the value 
of e itself. The symbol [5^ means 1x2x3x4x5. 

7. An angle a is usually supposed to be in radians. One radian is 
57 "30 degrees. 

Using the following series, calculate sin a and cos a to four significant 
figures. s .<> 7 

sina=a-.— + r^-pr + etc., 

cosa=l-^ + U-|-^+etc. 
Calculate also tan a, which is sin a^cos a. 



a 


sin a 


cos a 


tana 


Angle in degrees. 


0-001 


0-0010 


1-0000 


0-0010 


0-0573 


001 


0100 


0-9999 


0-0100 


0-573 


0-1 


0-0998 


0-9950 


01002 


5-73 


0-2 


0-1987 


0-9801 


0-2028 


11-46 


0-4 


0-3894 


#0-9207 


0-4228 


22-92 


0-6 


0-5646 


0-8253 


0-6843 


34-38 


1-0 


0-8416 


0-5403 


1-5577 


57-3 


1-5 


0-9975 


0-0706 


141589 


85-95 


2 


0-9094 


-0-4163 


-2-1842 


114-60 


3 


0-1411 


-0-9900 


-01423 


171-9 



Compare these values with those given in your tables. 
To find the sine, cosine, and tangent of angles greater than 90° refer 
to Art. 145. 



8. If ^ = 273 + ^ and <^=loge2=;^, ^"^ ^ ^^^ ^^^ following values of 6. 
The more correct formula is 

(^1 = 1 -0565 loge 273 + ^ ^ ^^~' ( 2^ ~ ^^^^ ) + 0"0902. 



42 ELEMENTARY PRACTICAL MATHEMATICS 

Calculate ^^ and see what error there is in the usual way of calculating. 



Given values 


t 


Answers. 


of d. 












<f> 


.1,1 





273 







50 


323 


•168 


•1683 


100 


373 


•3118 


•3134 


150 


423 


•4375 


•4415 


200 


473 


•5494 


•5570 


250 


523 


•6497 


•6628 



I usually employ the symbol ^«,, as the above <^ is the entropy of one 
pound of water at the temperature 6° C. 



9. If 



4>.=^os.4-,+''^-o-em 



and if j; = 273 + ^, 

find <^, for the following values of d. 



Given values of 6 





50 


100 


150 


200 


250 


4>s 


2 2^23 


r938 


r752 


16225 


1537 


1476 



(^1 is the entropy of 1 lb. of steam at the temperature d" C. 

10. The work done by a perfect Steam Engine, working on the 
Rankine cycle, is, per pound of steam, 



TF=140o{(^-g-^ologe/^ + x(l-^j)}, 



where 



X = 796^2-0^695/!. 



Note that t is the absolute temperature of the supply steam, t^ of 
the exhaust. 

If ?o=373, find W for each of the following values of t Find also 
w=1^98x 10^-^ TT in each case: this to is the weight of steam used 
per hour per indicated horse-power. 



Given t. 


w 


w 


600 


270,300 


7 33 


550 


231,200 


8^56 


500 


184,500 


1073 


450 


125,600 


15 76 


400 


50,220 


39 43 



11. If p is the pressure in pounds per sq. inch of saturated steam at 
the temperature 6° C, there is a useful empirical formula. 



ALGEBRA 

Find p in the following cases : 



43 



Given 6' 60 

1 


100 120 


140 


160 


170 


180 


190 


200 


Calculated p 2-88 


14-70 


•28-83 


52-52 


89-86 


115-1 


145-8 182-4 


225-9 


12. If e = 1 - r'*'"^, where y = 1*37, find e for the following values of r : 


r 


0-4 


0-3 


0-25 


0-2 017 


0-14 


012 


0-10 


Answers, e 


0-2876 


0-3594 


0-4013 


0-4487 


0-481 


0-5168 


0-5435 


0-5734 



e is the efficiency of the hypothetical gas or oil engine using the 
Otto cycle, r is the ratio of clearance to the greatest volume. 



13. If e = ] 


L-P-, 


where y- 


= 1-37, find € for the following values of P: 


P 


2 


6 


10 


14 


18 


22 


26 


Answers, e 


0-1708 


0-3835 


0-4630 


0-5095 


0-5419 


0-5660 


0-585 



e is the efficiency of the hypothetical gas or oil engine diagram using 
the Brayton cycle, where P is the pressure (in atmospheres) at which 
combustion takes place. 

14. Uniform beams of length I, supported at the ends and loaded with 
W in the middle ; / the moment of inertia of cross-section about the 
neutral axis ; E Young's modulus of elasticity ; the deflection D at the 
middle is 

D^ WPjASEI. 

I for a circular section of diameter d is %—- ; for a tube whose outside 

64 

diameter is c/q ^^^ inside d^ it is ^{dQ^ — d^) ; for a rectangular section 

of breadth h and depth d it is bd^/\2. For a rolled girder section it is 
^{bd^-{b-t){d-2tf} if b is breadth, d depth, and t is thickness of 
flanges and web. 

If W is the load which a beam of depth d will support at the centre, 
]V' = Slf/ld if /is the safe stress in the material ; d is the depth of the 
section. 

For wrought iron take ^=3x10^ lb. per square inch; /=10^ lb. 
per square inch. 

(1) If l = QO inches, rectangular section 6 = 1-2 inch, d=19 inch, find 
W, and when W is the load find the deflection. 

Ans. Tf' = 481-4 lb., Z) = 0-1053 in. 

(2) Circular section, diameter d=3 inches, 1 = 60 inches, find W, and 
when W is the load find the deflection. 

Ans. Tr = 1767 lb., D= 0*0667 in. 



44 ELEMENTARY PRACTICAL MATHEMATICS 

(3) Rolled girder section 1 = 350 inches, 6 = 6 inches, d=l6 inches, 
^=0'64 inch, find W\ and when W is the load find the deflection. 

Ans. If' = 8900 lb., /) = 0-4254 in. 

(4) Rectangular section, if rf=26 and l = 'SOd. If the load Tr=400 
produces a deflection 2) =0*1 inch, find d, and therefore b and l. 

Ans. d=l'8 inch, 6 = 0*9 inch, ^ = 54 inches. 

(5) Uniform beam of hollow circular section, thickness one-tenth of 
outside diameter, length 25 times outside diameter ; what is the diameter 
if 2000 lb. is the safe load ? What is the deflection when this load is upon 
the beam ? Ans. c?o = 4*643 inches, Z) = 0*1613 inch. 

(6) A beam of English oak 1 foot long, of rectangular section, 1 inch 
broad, 1 inch deep, supported at the ends, has a safe load W = 70 lb. at 
its middle and the deflection is then 0*021 inch ; find ^ and/ for English 
oak. Ans. B=l'Uxl(^,f=l260. 

15. Uniform beams of rectangular section, of breadth b and depth d 
and length I, loaded in the middle and supported at the ends ; the 
deflection under the load W is 

^ = 'bd-^ 

bd^ 
and the safe load Tf' =^-— , 

where c and g are constants for a particular material. 

For beams loaded uniformly and supported at the ends, D has '625 
times the above value and W has twice the above value. 

For beams loaded uniformly and fixed at the ends, I) has one-eighth 
of the above value, and W is 3 times as great. 

(1) The safe load W at the middle of a beam of English oak 1 inch 
broad, 1 inch deep, 12 inches long, supported at the ends is 70 lb. and its 
deflection D is 0*021 inch. Find the safe load and deflection for a beam 
of English oak 20 feet long, 8 inches broad, and 11 inches deep, 

Ans. If' = 3388 lb., 2) = 0*7635 inch. 

(2) If in the last case the beam is to be supported at the ends and 
loaded uniformly, find W and Z>. 

Ans. F' = 6776,i) = 0-4772 inch. 

(3) Instead of the case stated in (1) the beam is to be fixed at the 
ends and loaded uniformly, find W and B. 

A71S. Tf' = 10164 lb., Z)= 0-0954 inch. 

16. ^=6|sin3^, (4) 

^ = 6-7€-o=" sin 2*985^, (3) 

^=20^€-% (2) 

x=2l{€-'-€-^') (1) 

For the following values of ty calculate values of x from each of the 
four formulae, and tabulate as here shown. Remember that an angle is 
supposed to be specified in radians. 



ALGEBRA 



Values of t. 


X as calculated 


X as calculated 


X as calculated 


X as calculated 


from (1). 


from (2). 


from (3). 


from (4). 

















0-2 


1-634 


2-1952 


3-546 


3-7647 


0-4 


1-6086 


2-4088 


5-525 


6-21 


0-6 


1-3607 


1-9836 


5-461 


6-492 


0-8 


11213 


1-4515 


3-607 


4-5027 


10 


0-9195 


0-9956 


0-7729 


0-9393 


1-2 


0-7525 


0-6557 


-1-994 


-2-9520 


1-4 


0-6165 


0-4197 


-3-792 


-5-8133 


1-6 


0-5045 


0-2634 


-4-137 


-6-64 


1-8 


0-4133 


0-1609 


-3-082 


-5 1493 


20 


0-3382 


00971 


-1-135 


-1-86 


2-2 


0-2770 


0-0598 


0-9661 


2-08 


2-4 


0-2-267 


0-0358 


2-513 


5-2927 


2-6 


0-1856 


0-0213 


3-057 


6-0567 


4 


0-0459 


0-00049 


-1-182 


-3-572 



See note to exercise (21). When you plot these corresponding values 
of X and t on squared paper, you get four curves showing the motion of 
the same vibrating body with more and more damping. (See Art. 121. 



17. If x=a{<\>-s,m(^\ 

2/ = a{l-co8<f>). 

Take a = 10. For the following values of <^, calculate x and y and 
tabulate as here shown. ^ is in radians ; an angle of tt radians is 180° 
or two right angles. Later plot x and i/ of the curve which is called 
a cycloid. 



Values of <^. 


Values of x. 


Values of y. 











6 


0-236 


1-338 


IT 

4 


0-783 


2-929 


IT 

3 


1-81 


50 


TT 

2 


5-708 


100 


2ir 
3 


12-284 


15-0 


3ir 
4 


16-491 


1707* 


6 


21-18 


18-662 


IT 


31-416 


20-0 



See note to exercise (21). 



46 ELEMENTARY PRACTICAL MATHEMATICS 



18. If 



a; = a sin (pt + e). 



Let a = 10, jD = 37r, e = 77 or 30°. Calculate x for the following values 

of t, '06, -061, "062. See if your answers agree with the tabulated 
numbers of Art. 114. 

Now calculate ap cos{pt + e) for ^='0605 and -0615. 

Also calculate ap^sm{pt + e) for ^ = -061. 

Compare your answers with the numbers tabulated, Art. 114. 



19. If 



-li^-n 



if v = 100, r = l, Z=-01, calculate c for the following values of t, 0, -0001, 
•0002 ; -0010, -0011, -0012 ; '0100, 'OlOl, '0102. The true answers are 
tabulated in Ex. 10, Art. 99. 



20. Calculate c = 



\f^^i 



Ip 
sin {pt - e), where tan e = — ,\ia = lOO^p = 307r, 



r=l, l = -0\ for the following values of t, -0150, -0151, -0152. 
Also calculate v = asmpt for ^ = -0151. 
Compare your answers with the numbers tabulated in Ex. 11, Art. 99. 

21. y = as\n(hx + c). 

If tt = 10, 6 = 0-8727, find the value of y for each of the following values 

of X : V\ when c is ; 2°^, when c is -^ ; S'**, when c is - ; 4*'', when 

c is TT. Observe that the angle is in radians. 



Given values 




Ansvs^ers. 


Values of y. 




ofx. 


When c is 0. 


When c is 7r/6. 


When c is 7r/2. 


When c is tt. 




0-4 
0-8 
1-2 

s'-o 


0-0 
3 •4-20 
6-428 
8-660 

6-428 


5-000 

7-660 

9-397 

10-000 

9-397 


10-000 
9-397 
7-660 
5-000 

7-660 



- 3-420 
-6-428 
-8-660 

-6-428 



If you are not yet sure that you know how to find the sines of angles 
greater than 90°, work only those exercises in which the angle is less 
than 90°. After studying Art. 145, you can complete this exercise and 
draw the curve connecting y and x. 

22. If s = A sin (nt + e), where n^=g/Wh and ^ = 32-2. 

If A = -01, Tf=64-4, A = l, e = 0, calculate s for the following values of t : 

•0700, -0701, ^0702; "1400, •HOI, ^1402, 

use seven-figure tables. 

Compare your answers with the numbers tabulated, Ex. 12, Art. 99. 

23. When gas flows adiabatically from a vessel, where the pressure is 
Pq lb. per sq. ft. and its weight per cubic foot is Wq^ through an orifice 



ALGEBRA 



47 



into an atmosphere ; if we consider a stream tube which is of the small 
cross-sectional area A sq. feet at the place where the pressure is^; if v is 
the velocity at that place, g being 32*2 and y being r4 for air, 1"13 for 
dry or wet steam. 



:=2^a 



-A^--')' 



•(1) 



'o7 

where a is p/pQ ; and the weight of fluid per second in this stream being 
constant, we take 



A 






constant (2) 



It is not difficult to prove that dry or wet steam follows this law, as if 
it were a gas whose y = l"13. 

I give the answers for steam only. Take po = 100x144, ^^o = 0-23, 
y = l'13. Put (2) equal to some small constant, say 0*00073, and calculate 
V and A for the following values of p. 

Notice that A reaches a minimum value in the throat, and it can be 
shown that v there is the velocity which sound would have in fluid of the 
quality that exists there. A. false analogy due to the observation of 
the flow of liquid caused physicists and engineers to imagine that the 
velocity of a gas could never be greater than the velocity of sound. 
Notice that the stream tubes outside the orifice get larger in section, and 
the velocity is also larger. 



p-hl44 


A 


V 


100 Ve 


ry great 





90 


00732 


658 


80 


00541 


994 


70 


00489 


1245 


60 


00483 


1456 


57-85 


00481 


1512 


55 


00484 


1573 


50 


00488 


1708 


40 


00519 


1963 


30 


00589 


2252 


20 


00743 


2654 


15 


00889 


2910 


10 


01170 


3220 


5 


01430 


3506 


2-5 


03306 


4214 



I have assumed no friction, and there is practically no friction till the 
throat is reached, but in the diverging streams outside there is consider- 
able friction, which causes the calculated values of v to be too great. 



24. The formula 



is the value of h if the minus sign is taken, and it is the value of g if the 



48 ELEMENTARY PRACTICAL MATHEMATICS 

plus sign is taken. In the following five cases find h and g. Each of 
the five is a typical telephone or telegraph cable. 















Answers. 




« 


r 


I 


k 


< 






h 


9 


A 


5000 


88 





•05 X 10-6 





•105 


•105 


B 


5000 


18 


•0039 


■008x10-6 


10-6 


•0122 


•0302 


G 


5000 


2-97 


•0033 


•0096 X 10-6 


10-6 


•00281 


•0282 


D 


5000 


12 


•0010 


•0714 X 10-6 


5 X 10-6 


•0382 


•0564 


E 


60 


2-88 





•4095x10-6 





•005948 


•005948 



These answers are tabulated also in Ex. 9, Art. 149. 



25. In Exercise 24, a telephonic current 1 attenuates to €~^' in the 
distance x miles in each of the above cases, and it lags through the angle 
gx radians. Find the attenuation and the lag in degi'ees for each line in 
a distance of 50 miles. That is, calculate 

attenuation = g-^o^ and lag = bOg x 57 •S. 

Here are the answers : 





^ 


B 


c 


D 


E 


Attenuation 
Lag in degrees 


0052 
300° 


54 

87°^1 


0^868 
80° ^5 


0^14808 
16r'5 


743 
17°04 



26. Exercises on Electrical Conductors. 

(1) A round copper wire of length I and diameter d (both in centi- 
metres) has a resistance R ohms if 

R=lp^ld\ 

if /)= 1^7 X 10-6. Find R for one mile of wire of the following diameters. 
First show that /^ = 0•3482-^o^2. 



d 


1 


03162 


01 


0-03162 


001 


R 


3482 


3482 


34 82 


348-2 


3482 



These are the resistances when the currents are constant. 

(2) The above wires when conveying alternating currents of frequency 
ql'^TT undergo a fractional increase /5 of resistance if 

Show that ^ = 1 ^779 X 10" V^^*. 

For the above diameters, and at frequencies such that §' = 500, or 1000, 
or 2000, or 5000, or 10000, find the values of /?. 



ALGEBRA 



49 



Answers. Values of /i. 





Values of d. 




1 


0-3162 


0-1 


500 

1000 

2000 

5000 

10000 


•00445 
•01779 
•07116 
•4448 
P7790 


•00004 

•00018 

•0007 

•0044 

•0178 


•0000004 

•0000018 

•000007 

•000044 

•000178 



27. Loss in Induction Coils. 

(1) The watts tOe lost per cubic cm. in the iron by eddy currents are 

?«;e = 4-8/2lO-i2/52c;2 or 7-6/2lO-i^/?2«;2, 

if /= frequency, o?= diameter of each iron wire if the iron is in wires, 
if = thickness of iron sheet if the iron is in sheets. All dimensions are in 
centimetres, (i is the induction in c.g.s. units per sq. cm. 
The watts lOk, lost by hysteresis are 

^f;, = 2-5/;8inO-io. 

When /= 1000, fS = 2b, find the value of each of these. 

Ans. We = 0-003d'\ 2Ce=0-0047bf-, ?<';k = •0000431. 

(2) When c?=0-l, or 0-03162, or 0*01, calculate iCe/wn. 

Ans. 0-696, 0-0696, and 0-00696. 

(3) When ^ = 0^1, or 0^03162, or 0^01, calculate We/iOH. 

Ans. 1-102, 0-1102, and 0*01102. 
The iron is always thinner than ^ = 0*3 and the wire smaller than 
o?=0^03, so that hysteresis loss is much more important than eddy -current 
loss in all practical work. 

28. In any class of water turbine, if P is the total horse-power of the 
waterfall. If the height of the fall in feet, and 7i the revolutions per 
minute ; if R feet is the average radius of the place where water enters 
the wheel, then n = aH^''^P-'^/'^ and R = hP^!^H-^i\ where a and h are 
constants for a particular class of turbine. 

(1) From the manufacturer's list of a certain kind of low fall turbine 
I take one example, in which Zr=6, P=100, ?i = 50, /2 = 2^51. Find 
a and 6. ^tjs. a = 53*25, 6 = 0-963. 

We can now reproduce all the other figures of this manufacturer's list. 

(2) If ir=ll, P=80, find n and R. Ans. 7i= 119-6, i?= 1-425. 

(3) If ?i = 93 and H=}25, find P and R. Ans. P=181, R = \M7. 

(4) I find that for the Thomson turbine (called in America a Francis 
turbine), a = 22-75, 6=2-373. 

Find n and R if i/=250, P=74. Ans. w = 2628, /?=0-3247. 
Find Hi( R = 1-8A and P=95. A^is. i7= 29-22. 



P.M. 



50 ELEMENTARY PRACTICAL MATHEMATICS 



RATHER MORE ADVANCED GENERAL EXERCISES. 

1. If r is the radius of the earth, I the distance of the moon of mass 
M from the earth's centre, then M/{l + ry and Ml{l-ry are the 
accelerations towards the moon of points of the earth most remote ant 
nearest the moon. The tide-producing actions at these points are theii 
differences from the acceleration of the centre of the earth M/l^. Fine 
these in a simple approximate form, as r/l is very small. 

Ans. Let 7-/1 be called a, 

M M M(^ 1 \ M.^ .- _ .. 2J/ ^^r 



M M r 

In the same way^ ^-^— ^ - ^^ = ^^W 



Thus the tide-producing effect is inversely proportional to the cube oi 
the distance. The lunar tide of the earth is 2*1 times the solar tide, 
although the lunar attraction on the earth is very small compared with 
the solar attraction. 

2. The correct formula to use in a certain practical investigation is 

y = 0-75 + 2-59^2 _|. V0'5^'-i g|jj 3^,2 _,_ ^ j^g^ ^ _^ Q.3^^ ^y 

It was known that it would not be used for values of x less than "5 noi 
greater than \. 

It was quite impossible to work with it mathematically, so the 
following, found by means of squared paper, was used to replace it : 

y = 0-ll +3-833^ (2^ 

For various values of x compare the two formulae. 

3. In a certain armour-clad ship it is found that the easterly deviation 
D in degrees of the standard compass from the true magnetic north, when 
the ship's course makes the angle with the north of the standard 
compass {9 being measured clockwise looking downwards), is given by 
i)= 15 sin (^-1-25°) + 2 sin 2^ ; calculate D for sixteen values of B between 
0° and 360°, and plot on squared paper. Do this also for a certain 
wooden ship in which Z) = 5 sin 6 + 0'S sin W. 

If 9' is the angle between the ship's course and the true magnetic 
north, so that 9' =9 + 1), plot 9' and D in each of the above cases. 

4. On a Mercator's map, where a minute of longitude is always of the 
same length, a place in latitude I is at the distance 7n from the equator 

if m = log<. , — —. — y. (This is on the assumption that the earth is a 

TT * 1 - sm 6 ^ ^ 

sphere.) Calculate this for a few values of latitudes, and see that m is 

what sailors call " The Meridianal latitude " of a place. 

Close to the equator m is equal to the minutes of latitude. 

K T£ ^ *2r2 J '£ ha^l^/iT , 0*4769 ,, , 

5. If y = ~i=e~'''' and \i z= - ,, .-, , where a = — 7 — , compare the values 

of ?/ and z for many values of r from to oc . 

i", when A = l ; 2"", when /i = 10 ; 3"^, when /i = 100. 



CHAPTER VI. 
MENSURATION. 

29. Geometry ought to be practised not earlier than algebra. As 
soon as the student knows how to evaluate formulae he may go 
through some such course as the following, illustrating his study 
by practical geometry ; and assuming the truth of certain things 
from his own observation and his teacher's statements, he ought to 
prove the truth of other things which may not be so obviously true. 
There is a danger that a teacher will give a long useless course on 
practical geometry or a long course of abstract reasoning. What is 
wanted is a common-sense treatment of the subject such as will 
interest the average student, using the kind of reasoning that every 
thinking man employs in matters in which he is interested. Here 
is a sort of syllabus : 

The usual definitions. The measurement of angles in degrees, in 
radians, in right angles, and in revolutions. Draw any triangle, 
measure the angles, and add them together. Test the accuracy of 
your graphical work. Bisect any angle and any line ; draw per- 
pendiculars to lines. Construct triangles when given the three 
sides or two sides and the contained angle, or one side and the 
adjacent angles. Learn how to draw parallel lines. Get acquainted 
with the fact that a straight line drawn parallel to the base of a 
triangle divides the sides into proportionate segments. Prove that 
the bisector of an angle of a triangle divides the opposite side 
into segments proportional to the other sides. In equiangular 
triangles the sides are in the same proportions. Divide a straight 
line into parts proportional to those of a given divided line. The 
graphical methods of quickly dividing lines into many parts in 
certain proportions, by means of strips of squared tracing paper, 
and other labour-saving drawing office methods of working ought 



52 ELEMENTARY PRACTICAL MATHEMATICS 

to be given to students, not as things to be well remembered, but 
merely as illustrative exercises, for the wise student will load his 
memory with only a few important principles. It is most important 
that early in this work the student should draw a right-angled 
triangle, measure the sides, calculate the sine, cosine, tangent, etc., 
of the angles to compare with what the tables give, and he will have 
no difficulty in working simple problems on heights and distances, 
if he is not frightened by the information that this is a new and 
very advanced science called Trigonometry. 

There are only a few important principles which ought to be 
proved and illustrated and dwelt upon. 

Define similar plane and other figures. Similar areas are in 
proportion to the squares of their like dimensions. Similar volumes 
are in proportion to the cubes of their like dimensions. 

Prove that the area of a parallelogram is the length of a side 
multiplied by the perpendicular distance between this and the 
opposite side; that the area of a triangle is half the product of a 
side and the perpendicular on it from the opposite angle. Try ii 
this gives the same answer as half the product of the lengths of twc 
sides and the sine of the included angle. 

Show that in a parallelogram the complements of the parallelograms 
about the diagonal are equal. Prove the relations between the 
squares of the lengths of the sides of a right-angled triangle, and 
show that sin2^ + cos^^ = 1 . 

I am afraid that all this is frightfully unorthodox. The idea of 
replacing geometrical philosophy by arithmetical juggling is scorned 
by the modern mathematician. But who knows whether Euclid 
himself would not have used my method if he had not been so 
hopelessly ignorant of arithmetic. 

Show that the first ten propositions of the 2"'^ book of Euclid 
are identical with certain elementary algebraic statements, and the 
14th proposition is an extraction of a square root. Divide a number 
n into two parts x and n-x, so that n{n-x) = x'^. This is the IV^ 
proposition. Given two sides of a triangle and the contained angle, 
show how we find (1) the third side, (2) the other angles. 

The solution of triangles ought to be made as easy as possible, 
and there is no difficulty about this if teachers will only reflect on 
the great amount of unessential matter that is usually put before 
students. Hardly anybody nowadays needs any part of the un- 



MENSURATION 53 

essential (except for examination purposes) excrescences which have 
grown round the subject of the solution of triangles. They were 
never used except by a few surveyors, and even the surveyor who 
gets paid for making surveys nowadays almost never makes a 
survey, because he has his ordnance map. 

30. I have spoken of the first and second books of Euclid, so easy 
to make interesting to the average student, so stupefying as usually 
given. The fifth book is equivalent to two or three lines of the 

most elementary algebra, such as, if 7 = -,, then - = -, etc. I would 

impress upon a student the most general of all such propositions ; 

,, ..a G ^, ma + nb mc + 7id , ^ , , 

that II T =-7, then -, = j. whatever, m, n, p and n may be. 

b d' pa + qb pc + qd' j j 5 /- 1 j 

31. As for the third book of Euclid, I would simply illustrate 
propositions 1-19 by drawing, and assume the proofs to be un- 
necessary. On some of the following propositions there might be 
a little abstract reasoning, but not until the student has illustrated 
their truth by actual measurement. 

Prove that the angle at the centre is double the angle at the 
circumference when they have the same part of the circumference 
for their base, and that angles in the same segment are equal. The 
angle in a semicircle is a right angle. Also that the sum of the 
opposite angles of a quadrilateral inscribed in a circle is 180°. 
Also, the angle between a tangent and a chord is equal to the 
angle in one of the segments of the circle cut off by the chord. 
Prove that the products of the parts of chords of a circle cutting one 
another internally or externally are equal, and consider the special 
case of one of these being a tangent. 

In the fourth book of Euclid only one thing need be attended 
to — namely, how to 

Inscribe a circle in a given triangle. 

Of the 7"', 8'^ 9* and 10**^ books of Euclid I need not speak 
here, as they deal with arithmetic. Euclid did not, it is true, follow 
the mystical philosophy of numbers of Pythagoras and Plato, but 
these books are so utterly useless from every point of view that 
even the most orthodox of pedants allows them to be replaced by 
elementary arithmetic. There is some hope, therefore, of the rest 
of Euclid, and of all the recent pretences at reform of Euclid, 
disappearing from our school teaching in time. 



54 ELEMENTARY PRACTICAL MATHEMATICS 

32. Rules in Mensuration ought to be stated as formulae, and 
proved, if the proofs are easy, as part of the geometrical work. Surely 
it is an abominable thing to maintain the present artificial distinction 
between geometry and mensuration when they are both really the 
same, and to scorn arithmetic, which we now know, because Euclid, 
being ignorant, did not use it. The centre of area of a parallelogram 
is at the intersection of the diagonals. The centre of area of a 
triangle is one-third of the way along the bisector of a side towards 
the opposite angle. The circumference of a circle is 3*1 41 59c? or tt^^. 

The area of a circle is ^ ^^^ or ttt^. The area of an irregular polygon 

is found by dividing it into triangles whose areas may be added 
up. The area of a trapezium is half the sum of the parallel sides 
multiplied by the perpendicular distance between them. The area 
of the segment of the circle : — Find the area of the sector having 
the same arc (or J arc x radius, or ^rO^ if 9 (in radians) is the central 
angle subtended), and find the area of the triangle formed by the 
chord and the two radii, and subtract. 

There are some interesting rules which give the answer with a 
close approximation to accuracy, such as : 

Area of segment smaller than a semicircle. 
If h is the height and c is the chord, 

Area = ^ + %ch. 

Approximate Rule fm- s, the Length of a Circular Arc. s = ^(8l-L), 
where I is the chord of half the arc and L is the chord of the 
whole arc. 

We ought not to compel a student to keep in his memory more 
than a very few of the rules of mensuration. The most important 
thing is for a student, when he gets a problem to work, to be able 
to refer at once to some book in which the rules are clearly stated. 
In practical work a man may refer to any book for assistance ; if I 
had my way, every candidate at an examination would be allowed 
to bring into the examination room his favourite text-books. 

The 11*^ and 12*^ books of Euclid are now replaced by Descriptive 
Geometry and Mensuration. 

Projections of Points and Lines. Traces of Lines and Planes. Given 
two or three planes, by their traces find their intersections and 
the angles they make with one another. Find a plane containing 



MENSURATION 55 

a given line and point, or containing three given points. Find 
where a given line and plane meet and the angle between them. 
Find the angle between two given lines, etc., etc. The average 
student soon finds it easy to work simple exercises on cones and 
cylinders and their intersections. 



EXERCISES. 

1. A circular disc of copper is 3*2 inches in diameter. What is its 
circumference ? What is its area ? Ans. 10'05 inches, 8*04 sq, inches. 

2. The area of a circle is 20 sq. inches. What is its diameter ? What 
is its circumference? A^is. 5-04 inches, 15'8 inches. 

3. The parallel sides of a trapezium are 2*16 and 3'25 feet, their 
perpendicular distance apart is 1*57 feet. What is the area? 

Ans. 4*25 sq. feet. 

4. The segment of a circle has a chord of 8"56 inches and the height 
3'14 inches. What is the length of the chord of half the arc ? What is 
the area of the segment ? Ans. 5*31 inches, 19'77 sq. inches. 

5. The area of the curved surface of a right cylinder is the circum- 
ference multiplied by the length. What is the curved area of a cylinder 
4'5 inches diameter, 7 inches long? Aiis. 98*96 sq. inches. 

6. The curved surface of a right cone is the circumference of the base 
multiplied by ^ the slant height. 

The radius of the base is 2'6 inches ; the vertical height of the cone is 
5 '2 inches. What is the slant height ? What is the curved area ? 

A71S. 5*815 inches, 47*5 sq. inches. 

The student is supposed to know the shapes of prisms and cylinders 
bounded by parallel ends, and of cones and pyramids generally. 
The cubic content of a prism is always the area of the base multiplied 
by the perpendicular height, or what comes to the same thing, the 
area of cross-section by length along the axis. The axis of a prism 
goes through the centres of gravity of the ends. The centre of 
gravity of a prismatic body is half way along the axis. 

The cubic content of a pyramid or cone is found by multiplying 
the area of the base by one-third of the perpendiculai- height from 
base to vertex. The centre of gravity of a right cone is one quarter 
of the way along the axis from the base. The axis of such a body 
joins the centre of area of the base and the vertex. The curved 
area is easily constructed by practical geometry. The curved area 
of a right circular cone is half the circumference of the base 
multiplied by the slant side. 



56 ELEMENTARY PRACTICAL MATHEMATICS 

If A is the area of the base of the frustum of a cone, the area of 
the other parallel end being a, the height being h, the volume is h 
multiplied by ^(A -\-a + jAa). 

The volume of a ring is equal to the circumference of the circle 
which passes through the centres of area of the cross-sections, 
multiplied by the area of the cross-section. Thus, if R is the radius 
of the central circle and a is the radius of a circular section of a 
ring, the volume is 

F=2irEx7ra^ or 27r'^Ba^ (1) 

The proof of this rule is given in Art. 33. 

The area of a ring is the perimeter of a cross-section multiplied 
by the circumference of the circle which passes through the centres 
of all the perimeters. Thus, in the above ring of circular section 

the area ^ is A = i7rHla (2) 

The proof of this rule is given in Art. 33. 

Surface of a Sphere. Multiply the diameter by the circumference 
or take four times the area of a diametral circle, or S= rrd? or 47rr-. 

Curved Surface of a Bight Circular Cylinder. Multiply the circum- 
ference by the length, or S=27rrL. The curved surface of a 
spherical segment is equal to the curved surface of a cylinder of 
the same height as the segment, its base being a great circle of the 
sphere. 

Area of an Ellipse. Multiply the product of the major and minor 
axes by 7r/4. If the semi-axes are a and b, A= irab. 

Volume of a Sphere. —— or . 

Volume of a Plate. Area of plate multiplied by its thickness. 

Volume of the Segment of a Sphere. Subtract twice the height of 
the segment from three times the diameter of the sphere, multiply 
the remainder by the square of the height and by tt/G or '5236. 

The student ought to notice the importance in every case of 
writing out his rule as a formula. 

Fw example. A hollow circular cylinder, of external radius 7t, 
internal radius r, and length /, has a volume F, a curved surface aS^, 
and a weight W (if w is the weight per unit volume) : 

V=7r{R^-r^)l, 

S=2w(E + r)l, 

fF=^7rw(E^-r^)l 



MENSURATION 57 

Now, not only may we be asked to calculate V or *S' or JV^ but 
we may be given V or S oy W to calculate some of the dimensions. 

7. If /^ = 2, r = l, ^ = 5, find FandAS". Aiis. F= 47-12, >S'= 94-25. 

8. If F=160 cubic inches, 1 = 1 inches, r=2 inches. Ans. ^=3-358. 

9. If Jr=20 lb., ^ = 4 inches, r = 2-5 inches, ^(?=03 lb. per cubic inch, 
find I. Ans. ^ = 2-177. 

10. If F=230 cubic inches, »S=110 square inches, find R-r. 
Dividing the formula for F by the formula for ♦S', we get 

t^ = l{R-r\ so that /2 -r=4-182. 
Taking the formula for a ring given above, 

V=^Trmr\ A=4irmr. 

11. If 72 = 5, r = l, find Fand^. Ans. F= 98-68, ^ = 197-4. 
12 If A =230 cubic inches, F= 190 square inches, find r. 

Ans. -2-==i>*, so that r = 2x|§g or 1-652 inches. 
R is now found to be 3-526. 

13. If F= 120 cubic inches and ^=10 inches, find r. An^. r = -7798. 

14. A spherical shell, outside diameter 12 inches, weighs 100 lb. The 
material is such that 1 cubic inch weighs "26 lb. Find the internal 
diameter, which I here call .^7. 

(123 X -5236 - ^ X -5236) x -26 = 100, 
so that it is easy to find .r =9-977. 

15. A hollow cylinder of brass (0-3 lb. per cubic inch) is 11 inches long 
and 4 inches internal diameter, its weight is 40 lb. ; find its external 
diameter-, which I here call x. 

(^2 X -7854 - 42 X -7854) 1 1 x 0-3 = 40, 
so that it is easy to find ^ = 5-607. 

16. The rim of a cast-iron wheel (0-26 lb. per cubic inch) weighs 
20,000 lb, ; the rim has a rectangular section, thickness radially j;, size 
the other way I'b.v. The inside radius of the rim is IOjc. Find the actual 
sizes. Evidently 

(.^ X 1 -5^ X 27r X 10-5.^)0-26 = 20,000. 
Hence ^=9-194 inches. 

The following exercises are by intention unclassified : 

17. Find the area and circumference of a circle of 3-2 inches diameter. 

A71S. 8-041 square inches ; 10-05 inches. 

18. Find the diameter of a circle whose area is 15 square inches. 

Ans. 4-37 inches. 

19. Find the area of a triangle whose sides are 2 inches and 3-4 inches, 
the angle between being 74°. Ans. 3*268 square inches. 

20. Segment of a circle, chord 30 inches, height 4^ inches. Find the 
area. Ans. 91-35 square inches. 

If this were a parabolic segment, its area would be two-thirds of the 
chord multiplied by the height, or 90 square inches. 



58 ELEMENTARY PRACTICAL MATHEMATICS 

21. Find the area of the sector of a circle and the length of its arc, 
radius 6 inches, angle 50°. Ans. 15"71 square inches ; 5236 inches. 

22. Find the area and volume of a sphere of 10 inches radius. 

Ans. 1256"6 square inches ; 4180 cubic inches. 

23. Segment of sphere, height 6 inches, diameter of base 22'5 inches. 
Find volume. A71S. 1046 cubic inches. 

24. A right circular cone, radius of base 3"42 inches, height 6'42 inches ; 
find the area of its curved surface ; also find its volume. 

Ans. 78"14 square inches ; 78"63 cubic inches. 

25. An anchor ring has a mean radius of 4*2 inches, and the radius of 
a transverse circular section is 1*05 inch ; find the area of its surface, and 
also its volume. Ans. 174"1 square inches ; 91 '39 cubic inches. 

26. A hollow cylinder is 8*5 inches long ; its external and internal 
diameters are 5 inches and 3*5 inches ; find its volume and the area of its 
curved surface. Ans. 96*38 cubic inches ; 220'4 square inches. 

27. A piece of round copper wire 100 feet long weighs 4*3 lb. ; find its 
diameter. A cubic inch of copper weighs 0'32 lb. Ans. 0*119 inch. 

28. Find the area of the convex surface of a conical frustum 7*2 feet 
high, the radii of whose ends are 4*24 feet and 5*16 feet ; also find the 
volume. Ans. 214*3 square inches ; 501*3 cubic inches. 

29. Find the volume and weight of the rim of a cast-iron wheel of 
square section, the inside and outside diameters being 15 feet and*13 feet 
6 inches. Ans. 25*19 cubic feet ; 5*05 tons. 

30. The diameter of a circular cylinder is 4 feet ; find the area of the 
elliptic section made by a plane inclined to the axis of the cylinder at 50°. 
See Art. 36. Ans. 19*54 square feet. 

31. Find the area of an ellipse whose major and minor axes are 3*6 feet 
and 2*14 feet. What is the volume of a cone 2*5 feet high having this 
ellipse for its base ? Ans. 6*05 square feet ; 5*04 cubic feet. 

32. A spherical shell has inside and outside diameters of 6*5 and 
10*4 inches ; find its volume. A71S. 444 cubic inches. 

33. How many gallons of water are there in a tank 24 feet long, 15 feet 
wide, and 3 ft. 6 in. deep, when full ? Ans. 7875. See page 73. 

34. A pipe 3 metres long has external and internal diameters of 15 and 
11*5 centimetres ; find its weight in kilogrammes and pounds, if the 
specific gravity of the material is 7 '2. See page 73. 

A71S. 157*3 kilogrammes ; 346*8 pounds. 

33 Ring Theorems. Volume of a Ring. AB (Fig. 5) is any area ; 
if it revolves about an axis 00 in its own plane — the plane of the 
paper — it will generate a ring. The volume of this ring is equal to 
the area of AB multiplied by the circumference of the circle passed 
through by the centre of area of AB. Imagine an exceedingly 



MENSURATION 59 

small portion of the area a at the place P at the distance PQ = r 
from the axis. The volume of the elementary ring generated by 
this is a X 2irr, and the volume of the whole ring is the sum of all 




O Q O 

Fig. 5. 

such terms, and may be written as /^= lir^vr. But 2ar = RA if A is 
the whole area of AB, HE is the r of the centre of area * [or centre 
of gravity of the area, as many people call it]. Hence V= IttR x A^ 
the proposition to be proved. 

Area of a Ring. The area of the ring surface is the length of the 
perimeter or boundary s of AB multiplied by the circumference of 
the circle passed through by the centre of gravity of the boundary. 
Imagine a very short length of the boundary 8s at the distance r 
from the axis ; this generates a strip of area Ss x 27r?-. Hence the 
whole area is 27r2r . Ss, but ^r .8s = R .s if R is the distance of the 
centre of gravity of the boundary from the axis, and s is the whole 
perimeter. Hence the whole area of the ring is 5 x 'IrrR. 

Note that the centre of gravity of an area is not always the same 
as the centre of gravity of its boundary. 

Ex. 1. Find the volume and area of the rim of a fly wheel, its 
mean radius being 10 feet, its section being a square whose side is 
1-3 feet. Ans. F=l-32x27rx 10 = 106-2 cub. feet. 
Area = 4 x 1-3 x 27r x 10 = 326-8 sq. ft. 

Ex. 2. An ellipse, whose principal diameters are 10 and 6, 
revolves about an axis in its own plane ; the centre of the ellipse 
is at the distance 8 from the axis. What is the volume of the 



* This is the definition of centre of area. See Art. 54. 



60 ELEMENTARY PRACTICAL MATHEMATICS 

ring? Aqis. The area of the ellipse is 10 x 6 x '7854 = 47-124. This 
multiplied by ^tt x 8 is 2369, the volume. 

Ex. 3. At what distance x from the diameter is the centre of 
gravity of a semicircle? Eegard a sphere as a ring generated by 
the revolution of a semicircle about its diameter. The volume of 
the sphere is ^ttt^. The area of the semicircle is ^ttt"^. 



3 

so that X = ir/dir. 



*7rr3 = ^7r7-2 X 27ric, 



CHAPTER VII. 



ANGLES. 



34. An angle can be drawn : First, if we know its magnitude, 
in degrees; a right angle has 90 degrees. Second, if we know its 
magnitude, in radians ; a right angle contains 1'5708 radians. Two 
right angles contain 3-1416 radians. One radian is equal to 57*2958 
degrees. One radian has an arc BC equal in length to the radius AB 
or A C. It sometimes gets the clumsy 
name "a unit of circular measure." 
Third, we can draw an angle if we 
know either its siiie, cosine, or tangent, 
etc. Draw any angle BAC (Fig. 6). 
Take any point, P in AB, and draw 
PB at right angles to AC. Then 
measure FB, AP, and AB in inches 
and decimals of an inch. 

PB-^AP is called the sine of the 
angle. AB-^AP is called the cosine of the angle. PB-rAB 
called the tangent of the angle. 

Calculate these for any angle you may draw, and measure with 
a protractor the number of degrees in the angle. You will find 
from the Mathematical Tables whether your three answers are 
exactly the sine, or cosine, or tangent of the angle. The exercise 
will impress on your memory the meaning of the three terms. It 
will also impress upon us the fact that if we know the angle in 
degrees, we can find, by means of our tables, its sine, or cosine, or 
tangent ; and if we know any one of the sides AP or PB or AB of 
the right angled triangle APB and the angle A, we can find the 
other sides. Divide the number of degrees in an angle by 57*2958, 
and we find the number of radians. Suppose we know the number 




IS 



62 ELEMENTARY PRACTICAL MATHEMATICS 

of radians in the angle BAC, and we know the radius AB or AG, 

the arc BC is AB x number of radians in the angle. 

Given, then, a radius to find the arc, or given an arc to find the 
radius, are very easy problems. 

A student becomes accustomed, on seeing an angle drawn on 
paper, to judge from a mere glance how many degrees the angle 
contains. It would be an advantage to acquire the habit of judging 
how many radians there are in the angle. 

What we mean is, that he ought to be as ready to think in radians 
as in degrees, and to do this he requires to be familiar with the size 
of a radian. 

Ex. 1. Draw an angle of 35°. Find by measurement the sine, 
cosine, and tangent of the angle, and compare with the numbers in 
the tables. 

Calculate the number of radians ; that is, divide 35 by 57*296. 

Try if sin2 35° + cos2 35° = 1 ; if sin 35°^ cos 35° = tan 35° ; if 
tan^ 35° + 1 = sec.^ 35°, where sec. A means 1 -r cos A. 

Ex. 2. The sine of an angle is 0*25 ; find its cosine and tangent. 
Find the angle by actual drawing. How many radians % 

Ans. -9683, -2582, 14°-5, -2528. 

Ex. 3. What are the sine, tangent, and radians oi l^ degrees? 
Find them to four decimal places. Ans. Each '0262. 

Ex. 4. If in Fig. 6, A is 47° and AP is 5-23 feet, find AR and 
PB, Ans. AE= S-567, Pi^ = 3-824. 

Ex. 5. The arc BC (Fig. 6) is 12 feet, ACis 16 feet; find the 
angle. Calculate AP and PB ii AB is 10-b feet. 

Ans. -75 radian or 42° 58'; AP= 14-32 ft., Pi? = 9-779 ft. 

Ex. 6. The altitude of a tower observed from a point distant 
200 yards horizontally from its foot is 19°-3 ; find its height. 

Ans. 70-03 yards. 

Ex. 7. A mast consists of two parts, AB and BC. From a point 
in the horizontal from A, AB subtends 35° and AC subtends 45° ; 
find the ratio of AB to BC. Ans. 2-335. 

Ex. 8. From the top of a hill the angles of depression from the 
horizontal of two consecutive milestones in a line with the hill on a 
straight level road were found to be 12° and 6°-2 ; find the vertical 
height of the hill above the road. Ans. 0-2220 mile. 

Ex. 9. A point is in latitude 25°. If the earth were a sphere of 
3960 miles radius, how far is the point from the axis? What is 
the length of the circumference of the circle called a parallel of 
latitude that passes through the point? What is the 360*'' part 



ANGLES 63 

of this in length, and what is it called? What is the 360*^ part 

of a circle which is a meridian ? What is it called 1 

Ans. Distance from axis, 3589 miles ; circumference, 22,550 miles ; 

circum. 

— o^j— = 62'63 miles, the length of one degree of longitude ; 

— ^^ — = 69-10 miles, the length of one degree of latitude. 

Ex. 10. What is the length of a degree of longitude in latitude 
35°, taking the length on the equator as 60 nautical miles 1 

Ans. 49-12 miles. 

Ex. 11. The gunners' rule is that 1 inch at 100 yards subtends 
an angle of 1 minute. What is the percentage error of this rule ? 

Ans. 4-5 per cent. 

Ex. 12. A floating target is 20 feet in vertical height. A shell 
is descending at an angle with the horizontal of 4°. Within what 
limits of horizontal range will the shell hit the top or bottom of the 
target? Ans. 286 feet. 

35. Angular Velocity. If a wheel makes 90 turns per minute, 
this means that it makes 1 -5 turns per second. But in making one 
turn any radial line moves through the angle of 360 degrees, which 
is 6-2832 radians; so that 1-5 turns per second means 6-2832 x 1*5, 
or 9-4248 radians per second. This is the common scientific way in 
which the angular velocity of a wheel is measured — so many radians 
per second. 

If a wheel makes 30 turns per minute, its angular velocity is 
77 radians per second; n turns per minute mean ^irn radians per 
minute, or 27r?i-f-60 radians per second. One turn is the angular 
space traversed in one revolution. 

Ex. Show that the linear speed in feet per second of a point 
in a wheel is equal to the angular velocity of the wheel multiplied 
by the distance in feet of the point from the axis. 

Angular Acceleration. The Increase of Angular Velocity 7:>«r 
second. If a wheel starts from rest, and has an angular velocity 
of 1 radian per second at the end of the first second, its average 
angular acceleration during this time is 1 radian per second per 
second. 

Ex. 1. A shaft revolves at 800 revolutions per minute. What 
is its angular velocity in radians per second? An.i. 83-79. 

Ex. 2. A point is 3000 miles from the earth's axis and revolves 
once in 23 hours 56 minutes 4 seconds. What is its velocity in miles 
per hour? Ans. 787*5. 



64 ELEMENTARY PRACTICAL MATHEMATICS 

Ex. 3, A point is in latitude 58°. Take the earth to be a sphere 
of 4000 miles radius ; find the linear velocity of the point due 
to the earth's rotation. Ans. 556*4 miles per hour. 

Ex. 4. The average radius of the rim of a fly-wheel is 10 feet. 
When the wheel makes 150 revolutions per minute, what is the 
average velocity of the rim? Ans. 157*1 per second. 

Ex. 5. An acceleration of 1 turn per minute every second ; how 
much is this in radians per second per second? Ans. 0*1047. 

Ex. 6. A wheel is revolving at the rate of 90 turns a minute. 
What is its angular velocity in radians per second 1 

A point on the wheel is 6 feet from the axis ; what is its linear 
speed 1 If its distance from the centre be increased by 50 per cent., 
what does its speed become ] If, at the same time, the speed of the 
wheel increases 50 per cent., what is now the linear speed of the 
point 1 

Ans. 9*4*25 radians per second; 56*55 feet per second; 48*82 feet 
per second ; 127*2 feet per second. 

Ex. 7. There is a lever, OA, 30 inches long which works about 
an axis at 0. The lever is made to turn by applying a force at a 
point B in OA, 15 inches from 0, so that B receives a velocity of 
2 feet per second. What is the angular velocity of the lever 1 

If the same velocity had been given to the point A instead of B^ 
what would the angular velocity have been "? 

Ans. 1*6 radians per second; 0*8 radian per second. 

36. If a line AB makes an angle 6 with the horizontal, the 
projection of its length on the horizontal is A Boos $. 

Its projection on a vertical line is AB sin 6. 

If a plane area of A square inches is inclined at an angle 6 with 
the horizontal, its area as projected on the horizontal is A cos 6 
square inches. 

Try to prove that this must be so by dividing the area into strips 
by horizontal lines. 

Ex. 1. A plane area of 35 square feet is inclined at 20° to the 
horizontal; find its horizontal projection. Ans. 32*82 square feet. 

Ex. 2. The cross-section (a cross-section always means a section 
by a plane at right angles to the axis or line of centres of sections) 
of a cylinder is a circle of 0*7 inch radius. Find the areas of 
sections which make angles of 25° and 45° with the cross-section. 
Note that the cross-section is a projection of any other section. 

Ans. 1*699, 2*177 square inches. 

Ex. 3. The above cylinder is a tie bar of wrought iron. The 
total tensile load is 12,000 lb. How much is this per square inch 



ANGLES 65 

of the cross-section 1 How much is it per square inch of either of 
the other sections 1 A71S. 7794 lb., 7063 lb., 5512 lb. 

Ex. 4. The cross-section of a pipe is a circle of 15 inches 
diameter : what is the area in square feet 1 If 1 3 gallons flow per 
second, what is the velocity Fq^ AVhat is the area of a section at 
28° to the cross-section 1 What is the velocity F normal to this 
section, if normal velocity x area = cubic feet per second? Show 
that F is the resolved part of Fq in a direction normal to the 
section. Ans. 1-228, 1*7 feet per second; 1-39, 1*5 feet per second. 

Ex. 5. Part of a roof, shown in plan as 4000 square feet, is 
inclined at 24° to the horizontal. What is its area 1 

A71S. 4378'7 square feet. 

Ex. 6. A tie bar or short strut of 2 square inches cross-section ; 
what is the area of a section making 45° with the cross-section 1 
If the total tensile or compressive load is 20,000 lb., how much 
is this per square inch on each of the sections 1 Eesolve the total 
load normal to and tangential to the oblique section, and find how 
much it is per square inch each way. 

A71S. 2-828 square inches ; 10,000 lb , 7070 lb., 5000 lb. 

37. Vertical Line. A line showing the direction in which that 
force which we call the resultant force of gravity acts. It is a line 
at right angles to the surface of still water or mercury; 

Level Surface. A surface like that of a still lake, everywhere at 
the same level, and everywhere at right angles to the force of 
gravity or other volumetric force which is acting upon matter. It 
is not a plane surface. 

Curvature. For any curve we can find at any place what circle 
will best coincide with the curve just there. The radius of this 
circle is called the radius of curvature at the place. But since we 
say, for instance, that a railway line curves much, when we mean 
that the radius is small, the name curvature is always given to the 
reciprocal of the radius. Thus if the radius is 8 feet, we say that 
the curvature is \. If at another place the curvature is J, the 
change of curvature in going from the one place to the other is the 
difference between these two fractions. 

Example. In making 100 steps round a curve, my compass 
showing the direction of motion changes for N. to N.E. What is 
the average curvature? Ans. From N. to N.E. is 45 degrees or 
0-7854 radians, and this divided by 100 steps or -007854 radians 
per step is the average curvature. The reciprocal of this, or 127*3 
steps, is the radius of curvature, if the curvature is constant — that 
is, if the curve is an arc of a circle. 
P.M . E 



66 ELEMENTARY PRACTICAL MATHEMATICS 

Exercise. Through what angle must a rail 10 feet long be bent 
to fit a curve of half a mile radius 1 Ans. 0-22 degree. 



38. GENERAL EXERCISES. 

1. Find the angle subtended at the centre of a circle of radius 6y^ inches 
by an arc which is 1 inch long. Ans. 0"1572 radians or 9°. 

2. If an arc of 12 feet subtends an angle of 50°, what is the radius of 
the circle? ^?i^. 13-78 feet. 

3. If one of the acute angles of a right-angled triangle is 1*2 radians, 
what is the other ? Ans. 0*3708. 

4. A certain arc subtends an angle of \b radians if the radius of the 
circle is 2 "5 feet. Find the radius of the circle of which an arc equal in 
length to the first subtends an angle of 3'75 radians. Ans. 1 foot. 

5. Draw figures to show the following angles. Express them in radians : 
152°, 205°, -270°, 300°, -840°, 1350°. 

6. Draw a figure showing angles of 1, 2, 3, 4, 5 radians. 

7. A fly-wheel of 6 feet diameter on a shaft of 6 inches diameter 
revolves at 260 revolutions per minute. What is the speed of a point on 
the rim ? What is the speed of a point on the surface of the shaft ? 

Ans. 81*7 feet per second, 6*81 feet per second. 

8. The earth revolves about the sun, once a year, nearly in a circular 
path of 928 xlO^ miles radius. Find its speed in miles per second. 

Ans. 18 '5 miles per second. 

9. Assume that the earth revolves about its axis once in 24 hours (this 
is slightly wrong). Through what angle (in radians) does it revolve in 
one second ? Find the speed in feet per second of a point at the equator. 
Take the radius of the equator as 3963 miles. Ans. 0'0003, 1522. 

10. Find the speed relatively to the sun, in feet per second, of a point 
on the earth's equator, (i) at midday, (ii) at midnight. 

Ans. 99,232 feet per second at midday, 96,128 at midnight. 

11. A train is travelling in a curve of 0'5 mile radius, at the rate of 
20 miles per hour ; through what angle has it travelled in 10 seconds ? 

Ans. 0-111 radian or 6° -36. 

12. What angles do the large and small hands of a watch turn through 
between 11.15 a.m. and 2.30 p.m.? Ans. 1170°, 97°-5. 

13. The gunner's rule is that a halfpenny (the diameter of a halfpenny 
is one inch) subtends an angle of 1 minute at a distance of 100 yards. 
What is the percentage error in this rule? Aiu, 4'5 per cent. 

14. Assuming the earth to be a sphere of 8000 miles diameter, what is 
the circumference of the parallel of latitude 5r-5 ? The earth makes one 
revolution in 24 hours (nearly) ; what is the speed in miles per hour of 
South Kensington, which is in latitude 5r*5 ? 

Ans. 15,640 miles ; 652 miles per hour. 



ANGLES 67 

15. The inside of a hollow copper sphere is filled with water whose 
weight is 10 lb. ; what is the inside radius ? If the weight of the copper 
is 30 lb., what is its thickness ? A cubic inch of copper weighs 0*32 lb. 

Ans. 4*044 inches ; 0*414 inches. 

16. In a piece of coal there was found 0*1130 lb. of carbon, 0*0092 lb. 
of hydrogen, 0*0084 lb. of oxygen, 0*0056 lb. of nitrogen, 0*0071 lb. of ash. 
There being nothing else, find the percentage composition of the coal. 

Ans. 78*9, 6*4, 5*9, 3*9, 4*9 per cent. 

17. A pail is made in the shape of a frustum of a cone ; the internal 
diameters are 10 inches at the top and 7 inches at the bottom, height 
8 inches. Find its capacity and the weight of water which it will hold. 

Ans. 459*4 cubic inches : 16*54 lb. 



CHAPTER VIIL 
SPEED. 

39. I have given you exercises on velocity or speed. What do 
you mean by speed? When in a railway train we say that the 
speed is 30 miles per hour, what do we mean? Do we mean that 
we have gone 30 miles in the last hour, or that we are really going 
30 miles in the next hour? Certainly not. We may have only 
left the terminus 10 minutes ago ; there may be an accident in the 
next minute. It may be well to keep distances in feet and time in 
seconds. 

Find the time in seconds taken by a body to traverse, a certain 
distance measured in feet. This distance divided by the time is 
called the average velocity. Thus, if a railway train moves through 
200 feet in four seconds, its average velocity during this time is 
200 -r 4, or 50 feet per second. If we find with careful measuring 
instruments that it moves through 20 feet in "4 second, or through 
2 feet in -04 second, the velocity is 20 -v- *4, or 2 -^ -04, or 50 feet 
per second. It is important to remember that the velocity may be 
always changing during an interval of time, however short. To 
get the velocity at any instant we must make very exact measure- 
ments of the time taken to pass over a very short distance, and even 
this will only give us the average velocity during this short time. 
But if we make a number of measurements, using shorter and 
shorter periods of time, the average velocity becomes more and 
more nearly the velocity which we want. Thus, after 10 o'clock 
a man in a railway train making a careful measurement finds that 
the train passes over 200 feet in the next four seconds. He finds 
the average speed for four seconds after 10 o'clock to be 200 -f 4, 
or 50 feet per second. Another man finds that it passes over 
100*4 feet in the two seconds after 10 o'clock, and finds during 



SPEED 69 

these two seconds an average velocity of 100'4 -^ 2, or 50-2 feet per 
second. Another man finds 50-25 feet passed over in one second 
after 10 o'clock, which shows an average velocity of 50'25 feet per 
second. Another man finds 25-132 feet passed over in half a second 
after 10 o'clock, and finds 25-132 -^0*5, or 50-264 feet per second. 
Another man finds 12-567 feet in a quarter-second after 10 o'clock, 
and his observation gives 50*268 feet per second, and so on. It is 
evident that the values given by these various observations are 
approaching the real value of the velocity at 10 o'clock. Tabulating 
the results we have : 



Intervals of time in seconds 
after 10 o'clock. 


Average velocity in feet 
per second. 


4 

2. 
1 
h 


50-00 

50-20 

50-25 

50-264 

50-268 



Plot the two sets of numbers on squared paper, and draw a 
curve through the points so found. Produce the curve, and 
we have the means of finding the average velocity for an in- 
definitely small interval of time after 10 o'clock. This is the 
required velocity. 

The man who knows exactly what he means when he says "the 
speed of this train is 30 miles an hour," possesses the fundamental 
idea of the diflferential calculus, and can easily learn to use calculus 
methods. It is usually foolishly assumed that the idea cannot be 
possessed except by men who have devoted many years to mathe- 
matical study. 

It is known that a bullet falls freely vertically through the 
following intervals of time after two seconds from rest, at London. 
That is, in the time between 2 and 2-1, or in the time between 
2 and 2-01, or in the time between 2 and 2-001 seconds. I give 
here the distance fallen through : 



Intervals of time in seconds 


0-1 


0-01 


0-001 


Distances fallen through - 


6-601 


0-6456 


0-064416 


Average velocity^ 


66-01 


64-56 


64-416 



70 ELEMENTARY PRACTICAL MATHEMATICS 

We see that, as the interval of time after 2 seconds is taken less and 
less, the average velocity during the interval approaches more and 
more the true velocity at 2 seconds from rest, which is exactly 
64*4 feet per second. We see that even an interval of 0*001 second 
is too long; Ave ought to take the average velocity during a very 
much smaller interval than this if we wish to call it the real 
velocity at ^ = 2. 

Acceleration. This is the time rate of change of the velocity of a 
body. Thus it is known that the velocity of a body falling freely 
in London : 

At the end of one second is 32*2 feet per second, 
„ „ two seconds is 64*4 „ „ 

„ „ three „ „ 96-6 „ „ 

four „ „ 128-8 
and we see that there is an increase to the velocity of 32-2 every 
second. The acceleration in this case is always of the same amount, 
hence we call it uniform acceleration, and say it is 32*2 feet per second 
per second. 

EXERCISES. 

1. One mile per hour ; also, one knot. Convert each of these into feet 
per minute and feet per second. Ans. 88, 1*467 ; 101-3, 1-689. 

2. A destroyer travels at 32 knots. Convert this into English miles 
per hour. Ans. 36*85. 

3. Prove that 60 miles per hour means 00268 kilometres per second. 

4. Ten miles per hour. State this in feet per second, and in centi- 
metres per second. A7is. 14|, 447. 

5. An acceleration of 32*2 feet per second per second. State this in 
miles per hour per second, state it in centimetres per second per second. 

A71S. 21-95, 981-4. 

6. Two hundred gallons of water per minute. How many pounds 
per second ? How many cubic feet per second ? Ans. 33 3, 0-535. 

7. A round pipe 6 inches diameter has 30 gallons per second flowing 
through it. What is the velocity ? If the diameter becomes 10 inches, 
what is the velocity ? Calculate in the two cases the kinetic energy of 
one pound of water, this being the square of the velocity divided by 64-4. 

Ans. 24-5 feet per second, 8-8 feet per second, 9*3 foot-pounds, r2 foot- 
pounds. 

8. Two fine wires are 10 feet apart ; a bullet breaks them both. The 
breaking of each wire causes an electric spark to make a mark underneath 
a fixed platinum pointer on a revolving drum. If the drum is 4 feet in 



SPEED 71 

diameter, and revolves at 1000 revolutions per minute, and the spark- 
marks are found to be 1"32 feet asunder on the curved surface, assuming 
that the intervals of time between the breaking of the wires and making 
the marks were the same, find the time between the breaking of the wires 
and find the velocity of the bullet. 

The surface velocity is ^522^i^, or 209-44 feet per second ; 1-32 divided 

by this gives 0006303 second ; dividing this into 10 feet gives 1587 feet 
per second as the velocity of the bullet. 

9. In some gun experiments screens 150 feet apart were cut by a 
bullet at the following times (in seconds), counting from the time of 
cutting the first screen : 0, 0-0666, 0-1343, 0-2031, 0-2729, 0-3439,0-4159. 
Find the average velocity between every two successive screens. 

A71S. 2252, 2216, 2180, 2149, 2113, and 2083 feet per second. 

10. A body has passed through the space s feet, measured from some 
zero point in its path at the time t seconds, measured from some zero of 
time ; the law of the motion is 

5 = 12-2 -3-4^ + 6-7^2^ 

Calculate s when ^ is 4 ; calculate s one-tenth of a second later, when t 
is 4-1. Now find the distance passed through in this tenth of a second, 
and so find the average velocity. 

Thus s is 105-8 when t is 4, s is 110-887 when t is 4-1. 

[The student will notice that when working with a law supposed to be 
exact we may use many significant figures.] 

The space 5*087 feet being passed in 0-1 second, there is an avei'age 
velocity during this tenth of a second 

= -i- — = ^ , =50-87 feet per second, 
tmie 0-1 ^ 

Now let the student calculate s for ^ = 4-01, and for ^ = 4001, and find 
the average velocity for shorter and shorter intervals after t = 4, and so 
see what is the actual velocity at ^ = 4. 

If s and t are plotted on squared paper (see Art. 87), the slope of the 
curve is a measure of the velocity. 

40. It is inconvenient to refer to tables of numbers in bound 
books ; therefore the Board of Education sells the following tables 
almost for nothing (about one halfpenny for each copy), with the 
idea that the very poorest student ought to have copies. 

The exercise work of Chapters XXXIII. to XXXVI. will be 
made easier if students have tables of squares, square roots, and 
reciprocals. 

There are no published tables which are just what we want. 
Unnecessary tables such as logarithmic sines, etc., are a great nuisance. 
We need to work only to four significant figures. In the following 
list each of the tables needs two pages ; when will some enterprising 
publisher sell the collection for, say, a penny 1 



72 ELEMENTARY PRACTICAL MATHEMATICS 

Logarithms and antilogarithms ; Napierian logarithms and anti- 
logarithms ; sines, cosines, and tangents of angles from to 90° ; 
these angles ought to be in degrees and decimals of a degree, and 
make no reference to minutes. There ought to be a table such that, 
given tan 0, we find in degrees directly. A table to convert 
degrees into radians, and another to convert radians to degrees. 
Squares. Two tables of square roots. Keciprocals. These make 
14 tables, 28 pages. 



SPEED 73 



USEFUL CONSTANTS. 



1 mile = 1 '6093 kilometres. 

1 inch = 25*40 millimetres. i 4y / 

1 foot = 30-48 cm. ^^ W^l-^ . ^^f^^^f\\ ; 

J^]^qn^ -1604 cubic foot = 10 lb. of water at'^2° F. 

1 knot = 6080 feet per hour = l nautical mile per hour = l"15 miles per hour. 

Weight of 1 lb. in London = 445,000 dynes. 

One pound avoirdupois = 7000 grains = 453*6 grammes. 

1 kilogramme = 2-2046 lb. 

1 cubic foot of fresh water weighs 62'3 lb. 

1 cubic foot of air at 0° C. and 1 atmosphere weighs -0807 lb. 

1 cubic foot of hydrogen at 0° C. and 1 atmosphere weighs '(^559 lb. . 

1 foot-pound = 1 -3562 x 10^ ergs. 

1 horse-power-hour = 33000 x 60 foot-pounds = 2-6853 x 10^^ ergs. 

1 electrical unit = 1000 watt-hours = 1-34 horse-power-hours. 

1 Joule = 1 watt for 1 second =10" ergs. 

T 1 > • 1 i. 4. -^ r> li.) rr • r 774 ft.-lb. = 1 Fall. unit. 

Joules equivalent to suit Regnaults H is iiooQff n —in f- 

1 horse-power = 33000 foot-pounds per minute = 746 watts. 

1 watt = 10'' ergs per second. 

Volts X amperes = watts. 

1 atmosphere = 14*7 lb. per square inch = 2116 lb. per square foot = 760 mm. 

of mercury = 10^ dynes per sq. cm. nearly. 
A column of water 2*3 feet high corresponds to a pressure of 1 lb. per 

sq. inch. 
Absolute temp., t=^ C-f 273°, or (9° r-f-461°. 
7r = 3-l416. 

One radian = 57°-30 ; tt radians = 180°. 

To convert common into Napierian logarithms, multiply by 23026. 
The base of the Napierian logarithms is e = 2-7183. 
The value of g at London = 32I91 feet per sec. per sec. = 981 cm. per sec. 

per sec. 
Velocity of light, 3 x 10^" cm. per second. 
Mean density of the earth, 5-67 times that of water. 



ir^'^s'i.-^^nv^^ 



LOGARITHMS 








1 


2 


3 


4 


5 


6 


7 


8 


9 


12 3 4 


5 


6 7 8 9 


10 


0000 


0043 


0086 


0128 


0170 


0212 


0253 


0294 


0334 


0374 


4 9 13 17 
4 8 12 16 


21 
20 


26 30 34 38 
24 28 32 37 


11 


0414 


0453 


0492 


0531 


0569 


0607 


0645 


0682 


0719 


0755 


4 8 12 15 
4 7 11 15 


19 
19 


23 27 31 35 
22 26 30 33 


12 


0792 


0828 


0864 


0899 


0934 


0969 


1004 


1038 


1072 


1106 


3 7 11 14 
3 7 10 14 


18 
17 


21 25 28 32 
20 24 27 31 


13 


1139 


1173 


1206 


1239 


1271 


1303 


1335 


1367 


1399 


1430 


3 7 10 13 
3 7 10 12 


16 
16 


20 23 26 30 
19 22 25 29 


14 


1461 


1492 


1523 


1553 


1584 


1614 


1644 


1673 


1703 


1732 


3 6 9 12 
3 6 9 12 


15 
15 


18 21 24 28 
17 20 23 26 


15 


1761 


1790 


1818 


1847 


1875 


1903 


1931 


1959 


1987 


2014 


3 6 9 11 
3 5 8 11 


14 
14 


17 20 22 26 
16 19 22 25 


16 


2041 


2068 


2095 


2122 


2148 


2175 


2201 


2227 


2253 


2279 


3 5 8 11 
3 5 8 10 


14 
13 


16 19 22 24 
15 18 21 23 


17 


2304 


2330 


2355 


2380 


2405 


2430 


2455 


2480 


2504 


2529 


3 5 8 10 
2 5 7 10 


13 
12 


15 18 20 23 
15 17 19 22 


18 


2553 


2577 


2601 


2625 


2648 


2672 


2695 


2718 


2742 


2765 


2 5 7 9 
2 5 7 9 


12 
11 


14 16 19 21 
14 16 18 21 


19 


2788 


2810 


2833 


2856 


2878 


2900 


2923 


2945 


2967 


2989 


2 4 7 9 
2 4 6 8 


11 
11 


13 16 18 20 
13 15 17 19 


20 


3010 


3032 


3054 


3075 


3096 


3118 


3139 


3160 


3181 


3201 


2 4 6 8 


11 


13 15 17 19 


21 
22 
23 
24 


3222 
3424 
3617 
3802 


3243 
3444 
3636 
3820 


3263 
3464 
3655 
3838 


3284 
3483 
3674 
3856 


3304 
3502 
3692 
3874 


3324 
3522 
3711 
3892 


3345 
3541 
3729 
3909 


3365 
3560 
3747 
3927 


3385 
3579 
3766 
3945 


3404 
3598 
3784 
3962 


2 4 6 8 
2 4 6 8 
2 4 6 7 
2 4 5 7 


10 

10 

9 

9 


12 14 16 18 
12 14 15 17 
11 13 15 17 
11 12 14 16 


25 


3979 


3997 


4014 


4031 


4048 


4065 


4082 


4099 


4116 


4133 


2 3 5 7 


9 


10 12 14 15 


26 
27 
28 
29 


4150 
4314 
4472 
4624 


4166 
4330 

4487 
4639 


4183 
4346 
4502 
4654 


4200 
4362 
4518 
4669 


4216 
4378 
4533 
4683 


4232 
4393 
4548 
4698 


4249 
4409 
4564 
4713 


4265 
4425 
4579 

4728 


4281 
4440 
4594 
4742 


4298 
4456 
4609 
4757 


2 3 5 7 
2 3 5 6 
2 3 5 6 
13 4 6 


8 
8 
8 

7 


10 11 13 15 
9 11 13 14 
9 11 12 14 
9 10 12 13 


30 


4771 


4786 


4800 


4814 


4829 


4843 


4857 


4871 


4886 


4900 


13 4 6 


7 


9 10 11 13 


31 
32 
33 
34 


4914 
5051 
5185 
5315 


4928 
5065 
5198 
5328 


4942 
5079 
5211 
5340 


4955 
5092 
5224 
5353 


4969 
5105 
5237 
5366 


4983 
5119 
5250 
5378 


4997 
5132 
5263 
5391 


5011 
5145 
5276 
5403 


5024 
5159 
5289 
5416 


5038 
5172 
5302 
5428 


13 4 6 
13 4 5 
13 4 5 
13 4 5 


7 
7 
6 
6 


8 10 11 12 
8 9 11 12 
8 9 10 12 
8 9 10 11 


35 


5441 


5453 


5465 


5478 


5490 


5502 


5514 


5527 


5539 


5551 


12 4 5 


6 


7 9 10 11 


36 

37 

38 

9 


5563 
5682 
5798 
5911 


5575 
5694 
5809 
5922 


5587 
5705 
5821 
5933 


5599 
5717 
5832 
5944 


5611 
5729 
5843 
5955 


5623 
5740 
5855 
5966 


5635 
5752 
5866 
5977 


5647 
5763 
5877 
5988 


5658 
5775 
5888 
5999 


5670 
5786 
5899 
6010 


12 4 5 
12 3 5 
12 3 5 
12 3 4 


6 
6 
6 
5 


7 8 10 11 
7 8 9 10 
7 8 9 10 , 
7 8 9 10 


40 


6021 


6031 


6042 


6053 


6064 


6075 


6085 


6096 


6107 


6117 


12 3 4 


5 


6 8 9 10 


41 
42 
43 
44 


6128 
6232 
6335 
6435 


6138 
6243 
6345 
6444 


6149 
6253 
6355 
6454 


6160 
6263 
6365 
6464 


6170 
6274 
6375 
6474 


6180 

6284 
6385 
6484 


6191 
6294 
6395 
6493 


6201 
6304 
6405 
6503 


6212 
6314 
6415 
6513 


6222 
6325 
6425 
6522 


12 3 4 
12 3 4 
12 3 4 
12 3 4 


5 

5 
5 
5 


6 7 8 9 
6 7 8 9 
6 7 8 9 
6 7 8 9 


45 


6532 


6542 


6551 


6561 


6571 


6580 


6590 


6599 


6609 


6618 


12 3 4 


5 


6 7 8 9 


46 
47 
48 
49 


6628 
6721 
6812 
6902 


6637 
6730 
6821 
6911 


6646 
6739 
6830 
6920 


6656 
6749 
6839 
6928 


6665 
6758 
6848 
6937 


6675 
6767 
6857 
6946 


6684 
6776 
6866 
6955 


6693 
6785 
6875 
6964 


6702 6712 
6794 6803 
6884 6893 
6972 ; 6981 


12 3 4 
12 3 4 
12 3 4 
12 3 4 


5 16 7 7 8 
5 5 6 7 8 
4 5 6 7 8 
4 5 6 7 8 


50 


6990 


6998 


7007 


7016 


7024 


7033 


7042 


7050 


7059 7067 


12 3 3 


4 5 6 7 8 



The copyright of that portion of the above table which gives the logarithms of numbers from 1000 to 
2000 is the property of Messrs. Macmillan and Company, Limited, who, however, have authorised the use 
of the form in any reprint published for educational purposes. 



LOGARITHMS 








1 


2 


3 





5 


6 


7 


8 


9 


12345|6789 


51 
52 
53 
54 


7076 
7160 
7243 
7324 


7084 
7168 
7251 
7332 


7093 
7177 
7259 
7340 


7101 1 7110 
7185 1 7193 
7267 I 7275 
7348 7356 


7118 
7202 
7284 
7364 


7126 
7210 
7292 
7372 


7135 
7218 
7300 
7380 


7143 1 7152 
7226 1 7235 
7308 ' 7316 
7388 7396 


12334i5678 
12234J5677 
122345667 
12234 5667 


55 


7404 


7412 


7419 


7427 


7435 


7443 


7451 


7459 


7466 7474 


1223|4.5567 


56 
57 
58 
59 


7482 
7559 
7634 
7709 


7490 
7566 
7642 
7716 


7497 
7574 
7649 
7723 


7505 
7582 
7657 
7731 


7513 

7589 
7664 
7738 


7520 
7597 
7672 
7745 


7528 
7604 
7679 
7752 


7536 
7612 
7686 
7760 


7543 
7619 
7694 
7767 


7551 

7627 
7701 
7774 


122345567 
122345567 
11234 4567 
112344567 


60 


7782 


7789 


7796 


7803 


7810 


7818 


7825 


7832 


7839 


7846 


1123j4|4566 


61 
62 
63 
64 


7853 
7924 
7993 
8062 


7860 
7931 
8000 
8069 


7868 
7938 
8007 
8075 


7875 
7945 
8014 
8082 


7882 
7952 
8021 
8089 


7889 
7959 
8028 
8096 


7896 
7966 
8035 
8102 


7903 
7973 
8041 
8109 


7910 
7980 
8048 
8116 


7917 
7987 
8055 
8122 


1123 414566 
112334566 
1123i3!4556 
11233i4556 


65 


8129 


8136 


8142 


8149 


8156 


8162 


8169 


8176 


8182 


8189 


1123| 314556 


66 
67 
68 
69 


8195 
8261 
8325 
8388 


8202 
8267 
8331 
8395 


8209 
8274 
8338 
8401 


8215 
8280 
8344 
8407 


8222 
8287 
8351 
8414 


8228 
8293 
8357 
8420 


8235 
8299 
8363 
8426 


8241 
8306 
8370 
8432 


8248 
8312 
8376 
8439 


8254 
8319 
8382 
8445 


112 3 
112 3 
112 3 
112 2 


3 
3 
3 
3 


4 5 5 6 
4 5 5 6 
4 4 5 6 
4 4 5 6 


70 


8451 


8457 


8463 


8470 


8476 


8482 


8488 


8494 


8500 


8506 


112 2 


3 


4 4 5 6 


71 
72 
73 
74 


8513 
8573 
8633 
8692 


8519 
8579 
8639 
8698 


8525 
8585 
8645 
8704 


8531 
8591 
8651 
8710 


8537 
8597 
8657 
8716 


8543 
8603 
8663 

8722 


8549 
8609 
8669 
8727 


8555 
8615 
8675 
8733 


8561 
8621 
8681 
8739 


8567 
8627 
8686 
8745 


112 2 
112 2 
112 2 
112 2 


3 
3 
3 
3 


4 4 5 5 
4 4 5 5 
4 4 5 5 
4 4 5 5 


75 


8751 


8756 


8762 


8768 


8774 


8779 


8785 


8791 


8797 


8802 


112 2 


3 


3 4 5 5 


76 
77 
78 
79 


8808 
8865 
8921 
8976 


8814 
8871 
8927 
8982 


8820 
8876 
8932 
8987 


8825 
8882 
8938 
8993 


8831 
8887 
8943 
8998 


8837 
8893 
8949 
9004 


8842 
8899 
8954 
9009 


8848 
8904 
8960 
9015 


8854 
8910 
8965 
9020 


8859 
8915 
8971 
9025 


112 2 
112 2 
112 2 
112 2 


3 
3 
3 
3 


3 4 5 5 
3 4 4 5 
3 4 4 5 
3 4 4 5 


80 


9031 


9036 


9042 


9047 


9053 


9058 


9063 


9069 


9074 


9079 


112 2 


3 


3 4 4 5 


81 
82 
83 
84 


9085 
9138 
9191 
9243 


9090 
9143 
9196 
9248 


9096 
9149 
9201 
9253 


9101 
9154 
9206 
9258 


9106 
9159 
9212 
9263 


9112 
9165 
9217 
9269 


9117 
9170 
9222 
9274 


9122 
9175 
9227 
9279 


9128 
9180 
9232 

9284 


9133 
9186 
9238 
9289 


112 2 
112 2 
112 2 
112 2 


3 
3 
3 
3 


3 4 4 5 
3 4 4 5 
3 4 4 5 
3 4 4 5 


85 


9294 


9299 


9304 


9309 


9315 


9320 


9325 


9330 


9335 


9340 


112 2 


3 


3 4 4 5 


86 
87 
88 
89 


9345 
9395 
9445 
9494 


9350 
9400 
9450 
9499 


9355 
9405 
9455 
9504 


9360 
9410 
9460 
9509 


9365 
9415 
9465 
9513 


9370 
9420 
9469 
9518 


9375 i 9380 
9425 ; 9430 
9474 i 9479 
9523 j 9528 


9385 
9435 
9484 
9533 


9390 
9440 
9489 
9538 


112 2 
112 
112 
112 


3 
2 
2 
2 


3 4 4 5 
3 3 4 4 
3 3 4 4 
3 3 4 4 


90 


9542 


9547 


9552 


9557 


9562 


9566 


9571 


9576 


9581 9586 


1 1 2 1 2 


3 3 4 4 


91 
92 
93 
94 


9590 
9638 
9685 
9731 


9595 
9643 
9689 
9736 


9600 ! 9605 
9647 ; 9652 
9694 1 9699 
9741 9745 


9609 
9657 
9703 
9750 


9614 
9661 
9708 
9754 


9619 
9666 
9713 
9759 


9624 
9671 
9717 
9763 


9628 
9675 
9722 
9768 


9633 
9680 
9727 
9773 


0112)2 
112 2 
112 2 
1 1 2 1 2 


3 3 4 4 
3 3 4 4 
3 3 4 4 
3 3 4 4 


95 


9777 


9782 


9786 9791 


9795 


9800 


9805 


9809 


9814 


9818 


0112J23344 


96 
97 
98 
99 


9823 
9868 
9912 
9956 


9827 
9872 
9917 
9961 


9832 9836 
9877 ! 9881 
9921 , 9926 
9965 9969 


9841 
9886 
9930 
9974 


9845 
9890 
9934 

9978 


9850 
9894 
9939 
9983 


9854 
9899 
9943 
9987 


9859 
9903 
9948 
9991 


9863 
9908 
9952 
9996 


112 
112 
112 
112 


2 3 3 4 4 
2 3 3 4 4 
2 13 3 4 4 
2 3 3 3 4 



76 



ANTILOGARITHMS 








1 


2 


3 


4 


5 


6 


7 


8 


9 


12 3 4 


5 


6 7 8 9 J 


00 


1000 


1002 


1005 


1007 


1009 


1012 


1014 


1016 


1019 


1021 


001111222 


•01 
•02 
•03 
•04 


1023 
1047 
1072 
1096 


1026 
1050 
1074 
1099 


1028 
1052 
1076 
1102 


1030 
1054 
1079 
1104 


1033 
1057 
1081 
1107 


1035 
1059 
1084 
1109 


1038 
1062 
1086 
1112 


1040 
1064 
1089 
1114 


1042 
1067 
1091 
1117 


1045 
1069 
1094 
1119 


OOI1I1I1222 
001111222 
0011 111222 
0111 1;2222 


•05 


1122 


1125 j 1127 


1130 


1132 


1135 


1138 


1140 


1143 


1146 


011112222 


•06 
•07 
•08 
•09 


1148 
1175 
1202 
1230 


1151 
1178 
1205 
1233 


1153 
1180 
1208 
1236 


1156 
1183 
1211 
1239 


1159 
1186 
1213 
1242 


1161 
1189 
1216 
1245 


1164 
1191 
1219 
1247 


1167 
1194 
1222 
1250 


1169 
1197 
1225 
1253 


1172 
1199 
1227 
1256 


111 
111 
111 
111 


1 
1 
1 

1 


2 2 2 2 
2 2 2 2 
2 2 2 3 
2 2 2 3 


•10 


1259 


1262 


1265 


1268 


1271 


1274 


1276 


1279 


1282 


1285 


111 


1 


2 2 2 3 


•11 
•12 
•13 
•14 

•15 


1288 
1318 
1349 
1380 


1291 
1321 
1352 
1384 


1294 
1324 
1355 
1387 


1297 
1327 
1358 
1390 


1300 
1330 
1361 
1393 


1303 
1334 
1365 
1396 


1306 
1337 
1368 
1400 


1309 
1340 
1371 
1403 


1312 
1343 
1374 
1406 


1315 
1346 
1377 
1409 


111 
111 
111 
111 


2 
2 
2 
2 


2 2 2 3 
2 2 2 3 
2 2 3 3 
2 2 3 3 


1413 


1416 


1419 


1422 


1426 


1429 


1432 


1435 


1439 


1442 


111 


2 


2 2 3 3 


•16 
•17 
•18 
•19 


1445 
1479 
1514 
1549 


1449 
1483 
1517 
1552 


1452 
1486 
1521 
1556 


1455 
1489 
1524 
1560 


1459 
1493 
1528 
1563 


1462 
1496 
1531 
1567 


1466 
1500 
1535 
1570 


1469 
1503 
1538 
1574 


1472 
1507 
1542 

1578 


1476 
1510 
1545 
1581 


111 
111 
111 
111 


2 
2 
2 
2 


2 2 3 3 
2 2 3 3 
2 2 3 3 
2 3 3 3 


•20 


1585 


1589 


1592 


1596 


1600 


1603 


1607 


1611 


1614 


1618 


111 


2 


2 3 3 3 


•21 
•22 
•23 
•24 


1622 
1660 
1698 
1738 


1626 
1663 
1702 
1742 


1629 
1667 
1706 
1746 


1633 
1671 
1710 
1750 


1637 
1675 
1714 
1754 


1641 
1679 
1718 
1758 


1644 
1683 
1722 
1762 


1648 
1687 
1726 
1766 


1652 
1690 
1730 
1770 


1656 
1694 
1734 
1774 


112 
112 
112 
112 


2 
2 
2 
2 


2 3 3 3 
2 3 3 3 
2 3 3 4 
2 3 3 4 


•25 


1778 


1782 


1786 


1791 


1795 


1799 


1803 


1807 


1811 


1816 


112 


2 


2 3 3 4 


•26 
•27 
•28 
•29 


1820 
1862 
1905 
1950 


1824 
1866 
1910 
1954 


1828 
1871 
1914 
1959 


1832 
1875 
1919 
1963 


1837 
1879 
1923 
1968 


1841 

1884 
1928 
1972 


1845 
1888 
1932 
1977 


1849 
1892 
1936 
1982 


1854 
1897 
1941 
1986 


1858 
1901 
1945 
1991 


112 
112 
112 
112 


2 
2 
2 
2 


3 3 3 4 
3 3 3 4 
3 3 4 4 
3 3 4 4 


•30 


1995 


2000 


2004 


2009 


2014 


2018 


2023 


2028 


2032 


2037 


112 


2 


3 3 4 4 


•31 
•32 
•33 
•34 


2042 
2089 
2138 
2188 


2046 
2094 
2143 
2193 


2051 
2099 
2148 
2198 


2056 
2104 
2153 
2203 


2061 
2109 
2158 
2208 


2065 
2113 
2163 
2213 


2070 
2118 
2168 
2218 


2075 
2123 
2173 
2223 


2080 
2128 
2178 

2228 


2084 
2133 
2183 
2234 


112 
112 
112 
112 2 


2 
2 
2 
3 


3 3 4 4 
3 3 4 4 
3 3 4 4 
3 4 4 5 


•35 


2239 


2244 


2249 


2254 


2259 


2265 


2270 


2275 


2280 


2286 


112 2 


3 


3 4 4 5 


•36 
•37 
•38 
•39 


2291 
2344 
2399 
2455 


2296 
2350 
2404 
2460 


2301 
2355 
2410 
2466 


2307 
2360 
2415 
2472 


2312 
2366 
2421 
2477 


2317 

2371 
2427 
2483 


2323 

2377 
2432 
2489 


2328 
2382 
2438 
2495 


2333 
2388 
2443 
2500 


2339 
2393 
2449 
2506 


112 2 
112 2 
112 2 
112 2 


3 
3 
3 
3 


3 4 4 5 
3 4 4 5 
3 4 4 5 
3 4 5 5 


•40 


2512 


2518 


2523 


2529 


2535 


2541 


2547 


•2553 


2559 


2564 


112 2 


3 


4 4 5 5 


•41 
•42 
•43 
•44 


2570 
2630 
2692 
2754 


2576 
2636 
2698 
2761 


2582 
2642 
2704 
2767 


2588 
2649 
2710 
2773 


2594 
2655 
2716 
2780 


2600 
2661 
2723 
2786 


2606 
2667 
2729 
2793 


2612 
2673 
2735 
2799 


2618 
2679 
2742 
2805 


2624 
2685 

2748 
2812 


112 2 
112 2 
112 3 
112 3 


3 
3 
3 
3 


4 4 5 5 
4 4 5 6 
4 4 5 6 
4 4 5 6 


•45 


2818 


2825 


2831 


2838 


2844 


2851 


2858 


2864 


2871 


2877 


112 3 


3 4 5 5 6 


•46 
•47 
•48 
•49 


2884 
2951 
3020 
3090 


2891 
2958 
3027 
3097 


2897 
2965 
3034 
3105 


2904 
2972 
3041 
3112 


2911 
2979 
3048 
3119 


2917 
2985 
3055 
3126 


2924 
2992 
3062 
3133 


2931 
2999 
3069 
3141 


2938 
3006 
3076 
3148 


2944 
3013 
3083 
3155 


112 3 
112 3 
112 3 
112 3 


3 4 5 5 6 
3,4556 

4 1 4 5 6 6 
4 4 5 6 6 



ANTILOGARITHMS 








1 


2 


3 


^ 


5 


6 7 


8 


9 


12 3 4 


5 j 6 7 8 9 


•50 


3162 


3170 


3177 


3184 3192 


3199 


3206 j 3214 1 3221 3228 


112 3 


4 1 4 5 6 7 


•51 
•52 
•53 
•54 


3236 
3311 
3388 
3467 


3243 
3319 
3396 
3475 


3251 
3327 
3404 
3483 


3258 ' 3266 
3334 3342 
3412 3420 
3491 j 3499 


3273 
3350 
3428 
3508 


3281 3289 3296 
3357 3365 3373 
3436 3443 3451 
3516 i 3524 3532 


3304 
3381 
3459 
3540 


12 2 3 
12 2 3 
12 2 3 
12 2 3 


4 
4 
4 
4 


5 5 6 7 
5 5 6 7 

5 6 6 7 

6 6 6 7 


•55 


3548 


3556 


3565 


3573 3581 


3589 


3597 3606 3614 


3622 


12 2 3 


4 


5 6 7 7 


•56 
•57 
•58 
•59 


3631 
3715 
3802 
3890 


3639 
3724 
3811 
3899 


3648 
3733 
3819 
3908 


3656 
3741 
3828 
3917 


3664 
3750 
3837 
3926 


3673 
3758 
3846 
3936 


3681 
3767 
3855 
3945 


3690 
3776 
3864 
3954 


3698 
3784 
3873 
3963 


3707 
3793 
3882 
3972 


12 3 3 
12 3 3 
12 3 4 
12 3 4 


4 
4 

4 
5 


5 6 7 8 
5 6 7 8 
5 6 7 8 
5 6 7 8 


•60 


3981 


3990 


3999 


4009 


4018 


4027 


4036 


4046 


4055 


4064 


12 3 4 


5 


6 6 7 8 


•61 
•62 
•63 
•64 


4074 
4169 
4266 
4365 


4083 
4178 
4276 
4375 


4093 
4188 
4285 
4385 


4102 
4198 
4295 
4395 


4111 
4207 
4305 
4406 


4121 
4217 
4315 
4416 


4130 
4227 
4325 
4426 


4140 
4236 
4335 
4436 


4150 
4246 
4345 
4446 


4159 
4256 
4355 
4457 


12 3 4 
12 3 4 
12 3 4 
12 3 4 


5 
5 
5 
5 


6 7 8 9 
6 7 8 9 
6 7 8 9 
6 7 8 9 


•65 


4467 


4477 


4487 


4498 


4508 


4519 


4529 


4539 


4550 


4560 


12 3 4 


5 


6 7 8 9 


•66 
•67 
•68 
•69 


4571 
4677 
4786 
4898 


4581 
4688 
4797 
4909 


4592 
4699 
4808 
4920 


4603 
4710 
4819 
4932 


4613 
4721 
4831 
4943 


4624 
4732 
4842 
4955 


4634 
4742 
4853 
4966 


4645 
4753 
4864 
4977 


4656 
4764 
4875 
4989 


4667 
4775 
4887 
5000 


12 3 4 
12 3 4 
12 3 4 
12 3 5 


5 
5 
6 
6 


6 7 9 10 

7 8 9 10 
7 8 9 10 
7 8 9 10 


•70 


5012 


5023 


5035 


5047 


5058 


5070 


5082 


5093 


5105 


5117 


12 4 5 


6 


7 8 9 11 


•71 
•72 
•73 
•74 


5129 
5248 
5370 
5495 


5140 
5260 
5383 
5508 


5152 
5272 
5395 
5521 


5164 
5284 
5408 
5534 


5176 
5297 
5420 
5546 


5188 
5309 
5433 
5559 


5200 
5321 
5445 
5572 


5212 
5333 
5458 
5585 


5224 
5346 
5470 
5598 


5236 
5358 
5483 
5610 


12 4 5 

12 4 5 

13 4 5 
13 4 5 


6 
6 
6 
6 


7 8 10 11 

7 9 10 11 

8 9 10 11 
8 9 10 12 


•75 


5623 


5636 


5649 


5662 


5675 


5689 


5702 


5715 


5728 


5741 


13 4 5 


7 


8 9 10 12 


•76 

•77 
•78 
•79 


5754 
5888 
6026 
6166 


5768 
5902 
6039 
6180 


5781 
5916 
6053 
6194 


5794 
5929 
6067 
6209 


5808 
5943 
6081 
6223 


5821 
5957 
6095 
6237 


5834 
5970 
6109 
6252 


5848 
5984 
6124 
6266 


5861 
5998 
6138 
6281 


5875 
6012 
6152 
6295 


13 4 5 
13 4 5 
13 4 6 
13 4 6 


7 

7 
7 
7 


8 9 11 12 
8 10 11 12 

8 10 11. 13 

9 10 11 13 


•80 


6310 


6324 


6339 


6353 


6368 


6383 


6397 


6412 


6427 


6442 


13 4 6 


7 


9 10 12 13 


•81 
•82 
•83 
•84 


6457 
6607 
6761 
6918 


6471 
6622 
6776 
6934 


6486 
6637 
6792 
6950 


6501 
6653 
6808 
6966 


6516 
6668 
6823 
6982 


6531 
6683 
6839 
6998 


6546 
6699 
6855 
7015 


6561 
6714 
6871 
7031 


6577 
6730 
6887 
7047 


6592 
6745 
6902 
7063 


2 3 5 6 
2 3 5 6 
2 3 5 6 
2 3 5 6 


8 
8 
8 
8 

8 


9 11 12 14 

9 11 12 14 

9 11 13 14 

10 11 13 15 


•85 


7079 


7096 


7112 


7129 


7145 


7161 


7178 


7194 


7211 


7228 


2 3 5 7 


10 12 13 15 


•86 
•87 
•88 
•89 


7244 
7413 
7586 
7762 


7261 
7430 
7603 
7780 


7278 
7447 
7621 
7798 


7295 
7464 
763S 
7816 


7311 

7482 
7656 
7834 


7328 
7499 
7674 
7852 


7345 
7516 
7691 

7870 


7362 
7534 
7709 
7889 


7379 
7551 
7727 
7907 


7396 
7568 
7745 
7925 


2 3 5 7 
2 3 5 7 
2 4 5 7 
2 4 5 7 


8 
9 
9 
9 


10 12 13 15 

10 12 14 16 

11 12 14 16 
11 13 14 16 


•90 


7943 


7962 


7980 


7998 


8017 


8035 


8054 


8072 


8091 


8110 


2 4 6 7 19 


11 13 15 17 


•91 
•92 
•93 
•94 


8128 
8318 
8511 
8710 


8147 
8337 
8531 
8730 


8166 
8356 
8551 
8750 


8185 
8375 
8570 
8770 


8204 
8395 
8590 
8790 


8222 
8414 
8610 
8810 


8241 
8433 
8630 
8831 


8260 
8453 
8650 
8851 


8279 
8472 
8670 
8872 


8299 
8492 
8690 
8892 


2 4 6 8 9 
2 4 6 8 10 
2 4 6 8 10 
2 4 6 8 10 


11 13 15 17 

12 14 15 17 
12 14 16 18 
12 14 16 18 


•95 


8913 


8933 


8954 


8974 


8995 


9016 


9036 


9057 


9078 


9099 


2 4 6 8 10 


12 15 17 19 


•96 
•97 
•98 
•99 


9120 
9333 
9550 
9772 


9141 
9354 
9572 
9795 


9162 
9376 
9594 
9817 


9183 
9397 
9616 
9840 


9204 
9419 
9638 
9863 


9226 
9441 
9661 
9886 


9247 
9462 
9683 
9908 


9268 
9484 
9705 
9931 


9290 i 9311 
9506 9528 
9727 9750 
9954 9977 


2 4 6 8 11 
2 4 7 9 11 
2 4 7 9 11 
2 5 7 9 11 


13 15 17 19 
13 15 17 20 

13 16 18 20 

14 16 18 20 



ANGLES 



Angle. 


Chord. 


Sine. 


Tangent. 


Co-tangent. 


Cosine. 








De- 
grees. 


Radians. 















00 


1 


1414 


1^5708 


90° 


1 

2 
3 

4 


•0175 
•0349 
•0524 
•0698 


•017 
•035 
•052 
•070 


•0175 
•0349 
•0523 
•0698 


•0175 
•0349 
•0524 
•0699 


57^2900 
286363 
190811 
14-3007 


-9998 i 1-402 
•9994 1 1^389 
-9986 1377 
•9976 1364 


15533 
15359 
1^5184 
15010 


89 
88 
87 
86 


5 


•0873 


•087 


•0872 


•0875 


114301 


-9962 


1-351 


1^4835 


85 


6 

7 
8 
9 


•1047 
•1222 
•1396 
•1571 


•105 
•122 
•140 
•157 


•1045 
•1219 
•1392 
•1564 


•1051 
•1228 
•1405 
•1584 


95144 
8^1443 
71154 
6-3138 


-9945 
•9925 
•9903 
•9877 


1-338 
1325 
1312 
1^299 


14661 
1^4486 
1-4312 
1-4137 


84 
83 
82 
81 


10 


•1745 


•174 


•1736 


•1763 


5^6713 


•9848 


1^286 


1-3963 


80 


11 
12 
13 
14 


•1920 
•2094 
•2269 
•2443 


•192 
•209 
•226 
•244 


•1908 
•2079 
•2250 
•2419 


•1944 
•2126 
•2309 
•2493 


5^1446 
47046 
43315 
40108 


•9816 
•9781 
•9744 
•9703 


1^272 
1^259 
1^245 
1231 


1-3788 
1-3614 
13439 
1-3265 


79 

78 
77 
76 


15 


•2618 


•261 


•2588 


•2679 


37321 


•9659 


1218 


1-3090 


75 


16 
17 
18 
19 


•2793 
•2967 
•3142 
•3316 


•278 
•296 
•313 
•330 


•2756 
•2924 
•3090 
•3256 


•2867 
•3057 
•3249 
•3443 


3-4874 
3-2709 
3-0777 
2-9042 


•9613 
•9563 
•9511 
•9455 


1^204 
1190 
1176 
1161 


1-2915 
1-2741 
1-2566 
1-2392 


74 
73 
72 
71 


20 


•3491 


•347 


•3420 


•3640 


2^7475 


•9397 


1^147 


1-2217 


70 


21 
22 
23 
24 


•3665 
•3840 
•4014 
•4189 


•364 
•382 
•399 
•416 


•3584 
•3746 
•3907 
•4067 


•3839 
•4040 
•4245 
•4452 


2^6051 
2-4751 
2-3559 
2-2460 


•9336 
•9272 
•9205 
•9135 


1133 
1^118 
1-104 
1-089 


1-2043 
1-1868 
1-1694 
1-1519 


69 
68 
67 
66 


25 


•4363 


•433 


. -4226 


•4663 


2-1445 


•9063 


1-075 


11345 


65 


26 

27 
28 
29 


•4538 
•4712 
•4887 
•5061 


•450 
•467 
•484 
•501 


•4384 
•4540 
•4695 

•4848 


•4877 
•.5095 
•5317 
•5543 


20503 
1-9626 
1-8807 
1-8040 


-8988 
•8910 
•8829 
•8746 


1-060 
1045 
1-030 
1-015 


1^1170 
10996 
1-0821 
1-0647 


64 
63 
62 
61 


30 


•5236 


518 


•5000 


•5774 


1-7321 


•8660 


1000 


10472 


60 


31 
32 
33 
34 


•5411 
•5585 
•5760 
•5934 


•534 
•551 
■568 
•585 


•5150 
•5299 
•5446 
•5592 


•6009 
•6249 
•6494 
•6745 


1-6643 
1-6003 
1-5399 
1^4826 


•8572 
•8480 
-8387 
■8290 


-985 
-970 
•954 
•939 


1-0297 

1-0123 

-9948 

•9774 


59 
58 
57 
56 


35 


•6109 


•601 


•5736 


•7002 


1^4281 


•8192 


•923 


•9599 


55 


36 
37 
38 
39 


•6283 
•6458 
•6632 
•6807 


•618 
•635 
•651 
•668 


•5878 
•6018 
•6157 
•6293 


•7265 
•7536 
•7813 
•8098 


13764 
1-3270 
1-2799 
1-2349 


•8090 
•7986 
•7880 
-7771 


•908 
•892 
•877 
•861 


•9425 
-9250 
-9076 
-8901 


54 
53 
52 
51 


40 


•6981 


•684 1 •6428 


•8391 


11918 


-7660 


•845 


-8727 


50 


41 
42 
43 
44 


•7156 
•7330 
•7505 
•7679 


•700 i 'Q5Q1 
•717 i -6691 
•733 ! ^6820 
•749 -6947 


•8693 
•9004 
•9325 
•9657 


1-1504 
1-1106 
1-0724 
10355 


-7547 
•7431 
•7314 
•7193 


•829 
-813 
•797 

•781 


-8552 
-8378 
-8203 
-8029 


49 
48 
47 
46 


45° 


•7854 


•765 


•7071 


1-0000 


1-0000 


•7071 


•765 


-7854 


45° 








Cosine. 


Co-tangent. 


Tangent. 


Sine. 


Chord. 


Radians. 


Degrees. 


Angle. 



CHAPTER IX. 
USES OF SQUARED PAPER. 

41. A sheet of squared paper is covered with equidistant horizontal 
and vertical lines. Every tenth line is very distinct, so that it is 
easy to measure off horizontal and vertical distances without using 
a scale. The paper has its scales on it everywhere, in fact. 

Before 1876 sheets of squared paper were very expensive ; they 
were only used by a few people in important work. In that year 
Prof. Ayrton and I began to use it extensively in Japan, and when 
we returned to London and introduced at the Finsbury Technical 
College our methods of teaching Mathematics and Mechanical and 
Electrical Engineering and laboratory work, which have now become 
so common, we saw that one essential thing was the manufacture of 
cheap squared paper. It can now be bought for 6d. a quire instead 
of 8^. per sheet. Our students treat it almost like scribbling paper. 
The candidates in three important subjects of the Board of Education 
write their answers upon books of squared paper. It is of importance 
that the student should use many sheets of squared paper, use them 
lavishly. Formerly many men knew how squared paper might be 
used, but they really never used it, or if they did use it, they used 
it not for solving problems, but for illustrating methods of solving 
problems. 

I mean now to show you some of -the uses of squared paper. It 
would be easy to divide this subject up into 150 propositions and 
lead you on from one to the next, and so build up a science ; but 
here, at the very beginning, I want you to understand that, just as 
I said when describing the slide rule, all the following exercises are 
really one exercise. A student ought, after doing one or two of 
them, to see the general idea underlying them all, and if he will 
only practise by himself and exercise his common sense, he will be 



80 ELEMENTARY PRACTICAL MATHEMATICS 

able to solve any such problem, and furthermore, he will need no 
elaborate proofs ; things will be so self-evident as to require no 
proof. 

Proofs indeed ! Some people need proofs that they themselves 
exist. The mathematicians will tell you that this subject ought to 
be called Co-ordinate Geometry or Analytical Geometry. They 
will tell you that nobody ought to be allowed to begin it until he 
has mastered the most elaborate Algebraic and Trigonometric in- 
vestigations. Now I want you to understand that I have known it 
to be used, and used very wisely and well, by a man who could 
neither read nor write. 

42. I have here, cut from this morning's newspaper, this squared 
paper record of the rise and fall of the barometer and thermometer 
for the last week. Do you not all understand at once the meaning 
of this 1 Horizontal distances represent time since Sunday midnight. 
Vertical distances represent heights of the barometer in the one case, 
and heights of the thermometer in the other. 

43. Trades' newspapers have many records on squared paper. 
Here are some squared paper records taken from the engineering 
papers showing fluctuations in the prices of some of the metals. 

Here, again, are some curves showing the output of coal and of 
iron year by year since 1878 by Britain and America, Germany, 
France, Belgium, etc., which I find on exhibition in the museum. 
They show at a glance what you want to know. People interested 
in coal and iron will read an interesting story in every little 
fluctuation which you see. The general rate of growth of the 
industries is evident ; what is most striking being their enormous 
development in America. 

Think of a silk merchant in Yokohama putting on record in this 
way the price of silk per pound in Italy as it is telegraphed to him. 
Why does lie do it? First, he has a record of the price for years 
back ; a record read at a glanfee ; a record showing at a glance the 
times when the price reached a maximum or a minimum value; 
times when the market was disturbed. Second, he sees by the 
slope of the curve the rate of increase or fall in price. Third, if he 
plots other things on the same sheet of paper at the same dates, 
he will note what effect their rise and fall have upon the price of 
his silk. Fourth, he gets so much information from his curve that 
he is able to prophesy with more certainty than a man who has no 



USES OF SQUARED PAPER 



81 



such records ; indeed, he may actually be able to say with some 
certainty what price his silk will sell for in Italy a month hence, if 
he now sends a consignment. 

Practical money-making men and philosophers may use squared 
paper for so many useful purposes that I shall not attempt to 
enumerate them. 

I once read a clever article in the Nineteenth Century^ by one of 
our greatest statesmen, concerning the rates of increase of the 
English population and wealth. The reasoning was most abstruse. 
On taking the author's figures, however, and plotting them on squared 
paper, every result which he had reasoned out so elaborately was 
plain upon the curves, so that a boy could understand them. 

I was once sitting on a committee, when a manager was detailing 
reasons why attendance at certain classes was steadily dropping. 
In idleness, I manufactured some squared paper, quite roughly, 
and plotted the numbers, and it became at once evident that some 
curious event had happened at a particular date which had produced 
the mischief. This led at once to a rectification of the evil. Now 
I do not say that a man clever at figures would not have discovered 
this from the figures themselves, but the importance of the squared- 
paper method of working is that no worry over details of figures 
distracts one from the general story told by them. 

44. A student ought at once to use squared paper for himself, 
and use it with numbers in which he is interested. He will find 
such a book as Whitaker's Almanack very useful. 

You will see a table showing the price of Consols every year 
since 1790. Plot it as a curve, and get some notion in this way of 
the changing value of money. Or take this : 

Ex. 1. A certain insurance office gives assurance of £100 at 
death for the following yearly premiums : 



Age of insurer 


21 


25 


30 


35 


40 


45 


50 


55 


60 


Premium 


£ 8. d. 
2 3 1 


£ 8. d. 
2 6 6 


£ 8. d. 
2 11 9 


£ s. d. 
2 IS 2 


£ s. d. 
3 6 3 


£ s. d. 
3 16 4 


£ «. d. 
4 10 7 


£ s. d. 
5 13 8 


£ s. d. 
7 4 9 



You had better convert the shillings and pence into decimals of 
a pound. 

Thus take £5. 135. 8cL Here 8d. is j% or 0-667 of a shilling, and 
13-667 shillings = 0-6834 pound. So I wo^uld use £5-6834. [Perhaps 
P.M. F 



82 ELEMENTARY PRACTICAL MATHEMATICS 

you had better avoid exercises like this, which require tedious 
reduction to a decimal system.] 

Now, having drawn a curve through your plotted points, note 
that you can interpolate — that is, you can make a very good guess 
at the premium which would be charged an insurer of any inter- 
mediate age. Besides, the curve itself will teach you a good deal 
by its mere appearance. 

You might in the same way take an exercise from the table 
showing the price of an annuity for a person of a particular age. 

You will find an excellent exercise in the table showing the 
average length of life which may be expected by persons of a 
particular age. 

Plot the total amounts of gold or silver produced from mines 
every year since 1887. Draw the curve that lies most evenly 
among the points. See how nearly you can prophesy the amount 
of production during this coming year. 

Plot the total number of letters and of postcards posted every 
year since 1 885 in two curves. Note if one of them shows a peculiarity 
at any time whether the other was sympathetic. 

Plot the increasing revenue or expenditure in India, or the 
increasing commerce of the United States ; or the emigration from 
England ; or the value of imported mutton or apples every year 
since 1885. 

Ex. 2. The following are the numbers of half-time children in 
schools in twelve months following September 1, in each of the 
following years : 



Year - - - 1892 


1893 


1894 


1895 


1896 


Attendance 


164,018 


140,831 126,896 


119,747 


110,654 



Plot these and prophesy what the attendance will be in the 
following year. Also produce backwards and say what the probable 
attendance was from September 1st, 1891, to September 1st, 1892. 

[This exercise was given in one of my lectures in 1899, because 
just then there was a Bill before the House of Commons dealing 
with the half-time system in schools.] 

Plot the traffic receipts or the number of passengers of the 
Railways of the United Kingdom. 

Show in a curve how the number of second-class passengers has 
diminished since 1883, and the number of third-class passengers 
increased. 

I mention just a few exercises which one notices on picking up 
a book like Whitaker's Almanack. 



USES OF SQUARED PAPER 



83 



45. The following exercise was given in one of my lectures in 
1899, The tabulated numbers give some population statistics in 
millions ; let us say in the middle of each specified year : 



Year 


ISll 


1821 


1831 


1841 


1851 


1861 


1871 


1881 


1891 


England and\ 
Wales / 

Scotland 

Ireland - 


10-164 

1-806 


12000 

2-091 

6-802 


13-897 

2-364 
7-767 


15-914 

2-620 
8-175 


17-928 

2-889 
6-552 


20-066 

3 062 
5-779 


22-712 

3-360 
5-412 


25-974 

3-736 
5-175 


29-002 

4-026 
4-705 



Draw curves passing evenly through the plotted points. 

Do such exercises as these : (a) What was the probable population, 
in millions, of England and Wales in 1845? Ans. 16-72. (b) What 
was the probable rate of increase per annum in the middle of 
1845? A71S. 200,000 per annum, (c) What will be the probable 
population of England and Wales in 1901 and in 1911? Ans. 33-92 
and 38-58 millions. 

I give you this exercise (c) as it was given in 1899, because I now 
know the population in 1901 and 1911. In 1901 it was 32*5 and 
in 1911 it was 36-1. The above answers are therefore wrong, and 
this shows the danger of extrapolation. 

The student may here be troubled by my asking him for a rate of 
increase per annum at a particular time. He could give me the actual 
increase of population during the year 1845. The other idea is really 
familiar enough to him, but we must consider it in Chapter XVIII. 

In exercises like this, students will notice that although the num- 
bers may be perfectly exact, we may let our curve pass, not exactly 
through the points, but only evenly among them, if we are trying 
to see if there is some simple general law of growth of population. 

46. Exercise. The following measured numbers are taken from a 
certain table very useful to engineers. I want to know p when 
is 152. Also, I want to know when^ is 75. 



d 140 


145 


150 


155 


160 


165 


P 


52-52 


60-40 


69-21 


79 03 


89-86 


101-90 



By plotting on squared paper [and a student ought to use a large 
sheet ; some candidates in examinations waste a square inch of paper 
when they ought to waste a square foot of it], you can interpolate. 

Ans. e=l52,p = 7S-l; p = 75, 8=153. 

The student will notice that although the above numbers are 
derived from experiment, and are therefore probably slightly in 



84 ELEMENTARY PRACTICAL MATHEMATICS 



error, yet, as they have already been corrected for errors, his curve 
ought to go exactly through the plotted points.* 

In all the above work — when I ask a student to prophesy, what 
are the things that I have not warned him about, not told him 
about"? — I have not told him to exercise his common sense. I have 
not told him that in crossing the street he is in danger of being 
knocked down by a cab. Of course I could spend hours in talking 
about self-evident things ! But as for the philosophy of the 
idiosyncrasies of cab-drivers, it is too large a subject. 

47. Ex. 1. A student has made experiments and tabulates his 
observations as follows. Thus he found that when x was 80 his y 
was 0'55. Never mind now what his actual x or y was. His x was 
perhaps the temperature of steam and y was its pressure. Or his x 
may have been amperes of electric current and y may have been 
volts of potential difference between some two places. In any 
laboratory experiment we are always finding how one thing depends 
upon another ; x may be ampere turns on a magnet and y the 
magnetic field produced ; x may be the gallons of water per second 
flowing through a water meter and y may be the angular motion of 
a pointer on a dial, and we are going to graduate that dial from our 
experimental results. Anyhow, the student knows that there are 
errors in his measurements. He will plot the numbers on squared 
paper ; he will try to make some simple curve pass evenly among 
the points. Thus he will find the probable errors of his observations. 
If they seem to be too great he will probably reform his method of 
experimenting.! 



X 


80 


100 


150 


190 


250 


300 


y 


•55 


•78 


•97 


MO 


r22 


124. 



Now let a student take the above numbers and do as requested. 
He ought to know that he may use any scales whatever, and hence 

* Another way. Suppose we know that certain tabulated numbers follow a 
certain simple law without great error from one end of the table to the other, 
the law may be used for very exact interpolation. 

Example. The above values of d (temperature of steam) and p (pressure of 
steam in pounds per square inch) are known to follow this law sufficiently well 
for interpolation : p_^/^_(_jrj\5 

where a and h are constants. 

Given the following values, ^ = 1 50, ^ = 69 -21 ; ^ = 155, J9 = 79 '03, find a and h, 
and then calculate p when d is 152 ; also d when p is 75. 

[Exact methods of interpolating and finding rates of increase by tabulating 
differences need not be taken up here. See Art. 100.] 

t If he finds that they are greater than he thinks thej' ought to be, this is the 
beginning of a new kind of investigation, for it may be that his assumption 
that the curve is a simple one is wrong. 



USES OF SQUARED PAPER 85 

he had better use such scales as will give him points not merely in 
one corner of a sheet of paper. He will also notice that he may as 
well plot not the whole of y or of x^ for there are no values of y less 
than "55, nor of x less than 80. I make the probable errors of the 
above observed values of y to be respectively 

•013, -03, --03, -01, -02, --01. 
The - sign means that I think the observed y too small. 

Ex. 2. In the above case, y was not observed when x was 170; 
what is its probable, value ? I find 1*06. 

48. Ex. 1. A man makes saucepans. He has only made them 
of three sizes, as yet, but he knows that other sizes will be wanted. 
He wishes to publish a price list of many sizes. I don't know much 
about the saucepan trade, but suppose he has fixed on the following 
as really correct prices from every point of view : 

A 16 pint saucepan, price 87 pence. 

A 10 pint saucepan, price 68 pence. 

A 2 pint saucepan, price 28 pence. 

Now let him plot these sizes and prices on squared paper and join 

his points by a curve, say by bending a straight-edge. Any point 

on the curve shows size, and probable best price, of a saucepan. 

Thus I make the best price of a If gallon saucepan to be 81 pence, 
and of a 1 gallon saucepan to be 60 pence. 

Many manufacturers merely find carefully the prices of two sizes 
of a thing, and plotting these properly, they use a straight line as 
giving the price of all other sizes. 

Ex. 2. A man has made the following sizes of a certain type 
of small steam engine, and arranged their prices very carefully. 
Horse-power 4, price £44. 
Horse-power 12, price £108. 
Plot these on squared paper. Join the points by a straight line, 
and make out the probable prices of other sizes. 
Thus, I get, for 10 horse-power, the price £92. 
The student ought to examine a few price lists and find the rule 
on which the prices are calculated. 

49. When we are dealing with tables of numbers that are quite 
correct, our curve must pass exactly through the plotted points if we 
are going to interpolate. Thin battens of wood may be bent, weights 
resting on them here and there to enable a line to be drawn. Some- 
times a bent straight-edge is found to do well enough. The following 
is an exercise exceedingly interesting, not merely as illustrating in- 
terpolation, but as a continuation of what I have already said about 
logarithms. I quote the words used by me in a lecture in 1899. 



86 ELEMENTARY PRACTICAL MATHEMATICS 



To calculate a Table of Logarithms, or of Antilogarithms. 

Some friends of mine assert that no man or boy ought to be allowed 
to use logarithms until he knows how to calculate them. They say 
this, knowing that the calculation is a branch of Higher Mathematics, 
and that the average schoolboy after six years at mathematics finds 
it hopeless to even begin the study of the Exponential Theorem. It 
is a hard saying ! It is exactly like saying that a boy must not 
wear a watch or a pair of trousers until he is able to make a watch 
or a pair of trousers. It is the sort of unfeeling statement which 
so well illustrates the attitude of the superior person. 

Having recently discovered an easy way of calculating logarithms 
[I find that it is described in a book published about two days ago 
by Mr. Edser, an Associate of the Royal College of Science, and he 
has the priority over me as an inventor], and seeing that it is a very 
good illustration of our present work, I do not think that there can 
be any harm in giving it here. 

I assume that a boy can extract square roots by arithmetic. Let 
him, then, extract the square root of 10, and the square root of this 
again, and so on. Thus he finds 10^ = 10, 10* = 3^1623, 10^= 1-7783, 
10^=L3336, 10tV=M548, 10^V=i-0746. 

From these, by multiplication, he can find 10^"^, 10^^^, 10^^, etc. 

Thus iV ^s t^® logarithm of 1'1548 ; ^ is the logarithm of 1-3336. 

Let him use 0^125 instead of |^; in fact, let him use only decimals, 
and he has a table of which I give here only the beginning, the 
middle, and the end. 

If now he wants the logarithms of 
numbers between, say, 3 and o'i, let 
him plot three points on squared 
paper, using the whole of his sheet. 



Logarithm. 


Number. 




•03125 
•06250 
•09375 
•12500 


roooo 

r0746 
M548 
1-2409 
13336 


•46875 
•50000 
•53125 


2^9427 
31623 
3^3982 


•90625 

•93750 

•96875 

1-00000 


... 

8-0584 

8-6596 

9-3057 

10-0000 



Logarithm. 


Number. 


•46875 
•50000 
•53125 


2 9427 
3-1623 
3-3982 



Join these points by a curve; a 
slightly bent straight-edge enables 
this to be done very nicely. He will 
now be able to read off the logarithm 
of any number between 3 and 3-4, 
or the number for any logarithm 
between 0^47 and 0-53. In this way, 
even with cheap squared paper, he 
can calculate the tables correct to 4 figures. If he wants greater 

accuracy he must use 10^. 



USES OF SQUARED PAPER 



87 



50. Exercise. The simplest form of the exponential theorem as 
given in algebra is 

0*2 />*o /Tf»4 

- ^ th tO th 

e-=l+^ + - + ^+^ + etc., 

where e is the base of the Napierian system of logarithms. 

Calculate to five significant figures the values of e''^ e°-, e^'^, e°', 



Tabulate your answers. 



X 


e* 


«-^ 


0-1 


1-1052 


0-90481 


0-2 


1-2214 


0-81873 


1/3 


1-3956 


0-71654 


0-5 


1-6487 


0-60654 


2/3 


1-9477 


0-51343 


0-8 


2 "2255 


0-44934 


10 


2-7183 


0-36790 



by testing if e^"^ x e^ 



e^^'xe'i\ and 



Check your answers 
whether e -^ e"-^ = e" ^ 

Put y = e*. Then x = log^y. Plot x horizontally and y vertically 
on squared paper, and from your curve read off" the values of the 
Napierian logarithms of 1'5, 2-0, 2-5, and 2'7 as accurately as possible. 
Check your answers by converting to common logarithms and com- 
paring with your tables. [Multiply the Napierian logarithms by 
0-4343 or divide by 2-3026 to get the common logarithms.] 



y 


^OgeV 


logio2/ from the 
tables. 


1-5 
2-0 
2-5 
2-7 


0-407 
0-694 
0-9165 
0-994 


0-1761 
0-3010 
0-3979 
0-4314 



CHAPTER X. 



SOME MENSURATION EXERCISES. 



51. Before we take up the following exercises, it will be well to 
tell the student how we usually find the area of an irregular figure. 

If three equidistant ordinates to three points P, Q^ and E (Fig. 7), 
AP, BQ, and CE, are given, there are 
three approximate ways of finding the 
area APQECA. 

1 . Simpson's Rule assumes that the 
curve PQE is a parabola. 

(AP^iBQ + GE)^^ 

is the average ordinate, and this 

multiplied by ^C is the area.* 

*The following proof of Simpson's rule 
will not be understood by students until 
they have read Chap. XVIII. If 

z = a + hx + cx^, (1) 

and if Zj, Zg, and Zg are three values of 
z for equidistant values of a;, the average 
value of z between x^ and x^ is 



/>• 



c?a; = -(Zi + Z3 + 422). 



.(2) 




To prove this : 

Whatever z and x may be, we may repre- 
sent their relationship by means of a curve, which, if (1) is true, is a parabola 
with its axis parallel to the axis of z. It is of no consequence what zero we 
take in measuring x, for if we take x' = x + a, we find that the law connecting 
z and x' is exactly like (1), only that the constants are different. We can 
imagine, therefore, in this general proof, that iCg^O, Xx= -h, x^ = h. 

The average value of z is easily found to be 

a + '^ch^ (3) 

To find the values of a, 6, and c in any case (but we here do not need &), 
we insert in (1) the values iCi= -hy z = Zi ; 07.2 = 0, z = Z2; Xs = h, z = z.^, and we 
find a = Z2 and c = (Zj + Zg - 2z.2)/2h'^. 

Inserting these, we get the answer (2), which is Simpson's rule. 



SOME MENSURATION EXERCISES 89 

2. The Mid-ordinate Rule. Measure the mid-ordinates DS and 
ET. Take half their sum as the average ordinate of the whole 
curve. It is evident that this assumes the area of the curve to be 
the sum of the two rectangles AFGB and BHJC. 

3. The Trapezoidal Rule. (AF+CR + 2BQ)-^4: is the average 
ordinate. This assumes that the area is the sum of the areas of the 
two trapeziums APQB and BQRC as if the curve PSQ and the curve 
QTR were replaced by straight lines. 

When we get any three points of this kind with any kind of 
curve connecting them, it must be remembered that its appearance 
may be very different from what is shown in the figure. The curve 
in the figure is convex upwards ; the given curve may be concave 
upwards or may be partly concave and partly convex. 

Just as we find an area having equidistant ordinates, so we can 
find a volume if we have areas of equidistant parallel sections. 

For most of the simple solids, Simpson's rule gives the correct, 
and not merely an approximate answer. If A^ and A^ are the two 
parallel ends, and ^^ is the area of the section parallel to the ends, 
midway between them, then the average section is 

i(^i + ^3 + 4^2)- 

The volume of the frustum of a cone or pyramid ; the volume of a 
sphere or of a paraboloid of revolution between two parallel cross- 
sections; or, indeed, of any ellipsoid and other surfaces of the 
second degree ; the volume of the frustum of a wedge ; the volume 
of a prismoid — these can all be computed exactly by Simpson's rule. 

For example, a whole sphere may be said to have A^ = 0, ^3 = 0, 
and A^ = iri''^, so that the average section is J(0 + + ^irr^) or iirr^/G. 
This multiplied by 2r is the volume = l^^rr^. 

Definition of a Prismoid. Let there be two closed curves or 
irregular polygons on parallel planes, and let these be the ends of 
the prismoid. Imagine them joined by a surface which is made up 
of parts of cones or planes (a developable surface it is called). This 
is the most general definition of a prismoid that I can think of. 
Professor Harrison has proved that Simpson's rule is accurate for it. 

Ex. 1. The area of each end of a barrel is 8 square feet, the 
middle area is 10 square feet; what is the average area of cross- 
section of the barren Ans. (8 + 8 -t- 40) -^- 6 or 9-33 square feet. 
If the length of the barrel is 5 feet, its volume is 9*33 x 5 or 
46-67 cubic feet, by Simpson's rule. 



90 ELEMENTARY PRACTICAL MATHEMATICS 

Ex. 2. In a railway cutting, cross-sections 20 yards apart are 
91, 110, and 112 square yards in area; what is the total volume 
of earthwork? Ans. By Simpson's rule, the average section is 
<91 + 112 + 440) -^ 6 or 107-17 square yards; the volume is 107*17 x 40 
or 4287 cubic yards. 

Ex. 3. The frustum of a right cone has a circular base 4 inches 
diameter, a circular top 2*5 inches diameter parallel to the base ; 
the perpendicular distance between base and top is 5 inches ; what 
is its volume 1 In this case Simpson's rule gives the correct answer. 
Ans. The mid area evidently has a diameter of |^(4 + 2'5) or 
3-25 inches. The three areas are the squares of the diameters multi- 
plied by -7854, so that the average area of section is 
•7854(16 + 6-25 -f 42-25) -^ 6 or 8-443. 
The volume is therefore 8-443 x 5 or 42*22 cubic inches. 

Ex. 4. A reservoir, whose sides are planes, is at the top a 
rectangle, 600 feet by 100 feet; at the bottom it is a rectangle, 
whose corresponding sides are 200 feet and 70 feet; the vertical 
depth is 50 feet. What is the volume 1 The mid section is 
evidently a rectangle whose sides are ^(600 + 200) and |(100-i-70) 
or 400 and 85 feet. Therefore, ^^ = 600 x 100 or 60,000 square feet, 
^2 = 400 X 85 or 34,000, ^3 = 200 x 70 or 14,000. The average section 
is 1(60,000 + 14,000 + 136,000) or 35,000 sq. feet, and the volume is 
35,000 X 50= 1,750,000 cubic feet. 

52. We may divide an irregular area up into many parts. The 
following rules may be at once derived from those given above. I 
do not give the trapezoidal rule, but it is easily found if wanted. 

Simpson's Rule. Divide the area into any even number of parts 
by an odd number of equidistant parallel lines, or ordinates, the 
first and last being possibly of no length, for they must touch 
the boundary curve. Take the sum of the extreme ordinates (in 
many cases 0), four times the sum of the even ordinates, and twice 
the sum of the odd ones (omitting the first and last) ; multiply the 
total sum by one-third of the common distance asunder. This will 
give the area nearly. 

The mid-ordinate rule is : divide the area into any number of 
parts by equidistant parallel lines, the first and last touching the 
bounding curve. Midway between every two, measure the breadth 
of the figure ; add up these breadths and divide by the number of 
them; call this the average breadth. Multiply by the length or 
perpendicular distance between the extreme lines to get the area. 
In indicator diagram work we usually take ten parts. 



SOME MENSURATION EXERCISES 



91 



The student ought to know the sort of error he may expect in 
using such methods. Let him draw a circle and divide it up ; let 
him use Simpson's rule to find its area ; he will find a considerable 
discrepance from the correct answer. This is a particularly bad 
case for the Simpson's rule. He ought to compare the two methods 
in other cases. 

I prefer to use the mid-ordinate rule myself. I have often 
wondered why the Simpson's rule (extended to many ordinates) 
was so often mentioned (and always by people who never need to 
compute an irregular area), because it is not easy to remember. I 
have come to the conclusion that it is because it is arrived at 
through the use of the integral calculus, and it looks learned to 
speak about it. 

A planimeter is an instrument which enables us to measure 
the area of a figure in square inches or square centimetres or 
other units. It is very valuable when one has many areas to 
find. We let the tracing point go round the boundary of our 
area exactly to the starting point ; we can start anywhere ; 
the increase in the dial reading gives the area. The principle of 
the Amsler planimeter is not difficult to understand, and ought 
to be given to students. When a curve has many loops, there 
is an interesting rule as to whether the area of a particular 
part ought to be called positive or negative. We need pay no 
attention to any such rule when working with the planimeter. 
Hence it is that the planimeter is so valuable when we require the 
average pressure on a gas or oil engine indicator diagram. We 
find the area by the planimeter, and divide by its extreme length 
parallel to the atmospheric line. 

53. Just as we find an area having equidistant ordinates, so we 
can find a volume if we have areas of equidistant parallel sections, 
using either Simpson's rule or the other rule. 

Ex. 1. A reservoir with irregular sides has the following 
dimensions. When filled with water to the vertical height h feet 
above a datum level, the following is the area A square feet of the 
water surface. The height of the lowest point above datum level 



h ! 20 

1 


25 


30 


35 


40 


45 


50 


55 


60 


65 


70 


A 





11,800 


23,600 


37,100 


51.000 


61,500 


76,010 


89,000 


102,000 


118,250 


130,300 



92 ELEMENTARY PRACTICAL MATHEMATICS 



Find the volume in cubic feet when filled up to h = 70. 

Ans. 3-177x106. 
What is the volume between h = 44^ and h = 45|. 

Ans. 61,500 cubic feet. 

Ex. 2, A series of soundings taken across a river channel is 

given by the following table, x feet being the distance from one 

shore, and y feet the corresponding depth. Find {a) the area of 

the section, (h) the position of its centre of gravity. (See Art. 54.) 



X 





10 


16 


23 


30 


38 


43 


50 


55 


60 


70 


75 


80 


y 


10 


20 


26 


28 


30 


31 


28 


23 


15 


12 


8 


6 






Ans. (a) 1583 sq. ft. ; {b) ir = 34-l ft., ^=12-1 ft 

(c) If the average speed of the water normal to the cross-section 
is 4-5 feet per second, what is the quantity in cubic feet per second 
flowing^ Ans. 7123 cub. ft. per sec. 

(d) If there is an available fall of 10 feet, what is the horse- 
power? Ans. 8069. 

Note. — A cubic foot of water weighs 62*3 lb. ; if iv lb. per minute can fall 
vertically h feet, the horse-power available is tvh -^ 33,()00. 

Ex. 3. Draw a figure something like a steam-engine indicator 
diagram, of length about four inches, and greatest height 3 inches. 
Find its area by means of a planimeter, after testing the accuracy of 
the planimeter by finding the area of a square. 

Now compute the area by dividing it into a few strips, and then 
into many strips, using the Simpson's rule and also the mid-ordinate 
rule. Compare your answers, and thus get an idea of the relative 
accuracies of various methods. 

Ex. 4. Describe a circle of, say, 6 inches diameter. Draw the 
diameter; divide it into six equal parts, and draw ordinates and 
measure them. Compute the area by Simpson's rule and then by 
the mid-ordinate rule. The true answer is of course 28-274 square 
inches. You will find that the Simpson's rule is not very satisfactory 
in this case. Now divide into 12 equal parts and repeat your work. 
Both answers will be more nearly right, but still the Simpson's 
method is the more inaccurate. 

Ex. 5. The water plane of a vessel is 200 feet long ; the central 
line is divided into 20 equal parts and the breadths at each place in 
feet are 0, 22, 27, 29, 30, 30-5, 30-5, 30*5, 30-5, 30, 29-5, 28, 26*2, 
24-5, 21-5, 18, 14-5, 11, 7-1, 3-9, 0. Find its area by Simpson's 
rule. If the draught of the ship lessens by 1 inch in salt water, 
what is the loss of tonnage 1 Ans. 4477 sq. feet; 10-66 tons. 

Ex. 6. The internal area of each of the two ends of a barrel is 
12*35 sq. feet; the area of the middle section is 14-16 sq. ft. ; the 
axial length of the barrel is 5 feet. What is its volume? What 
weight of water will it hold? Ans. 67-78 cub. ft. ; 4223 lb. 



CHAPTER XI. 
MENSURATION EXERCISES. 

54. Centres of Gravity. We often speak of the centre of gravity 
of a body, or of an area, or of a curve, when we mean centre of 
inertia, or centre of area, or linear centre. 

1. Multiply each small portion of mass m of a body by its 
distance x from a plane ; indicate the sum as 2wx. Divide this by 
the whole mass of the body M or Im, and we have the distance of 
the centre of inertia or mass of the body from the same plane. 

Make this calculation for three planes, and you get the exact 
position of the centre of mass. 

If a body is symmetrical about a plane, one-third of the labour 
is saved. If the body is symmetrical about an axis, two-thirds of 
the labour is saved. 

We write our rule as '^mx = Mx. 

2. Multiply each small portion a of an area by its distance x from 
a straight line in the same plane. Indicate the sum as 2a.r. Divide 
this by the whole area A or 2a, and we have the distance of the 
centre of area from the line. Make this calculation for two lines 
not parallel, and you get the exact position of the centre of area. 

Ex. 1. Let PBGC (Fig. 8) be an irregular figure. To find its 
area and the position of its centre of area, draw FD and (9(7, two 
parallel lines touching the extreme boundary, and let DAO be any 
line at right angles to these. Divide DO into many equal parts, and 
draw ordinates at the middle of each. 

The sum of all such ordinates as BC divided by the number of 
them is the average breadth, and this multiplied by DO is the area. 
Or we may write it, if d is the distance AQ oi the ordinates, 
asunder : 

Area A=d(JII + JK+ LM + NP + BC + etc.). 



94 ELEMENTARY PRACTICAL MATHEMATICS 

Also, if OX is the horizontal distance of the centre of gravity of 
the area from OG, by the definition 

d.HIxU-\-cl.JKxnd + cl.LM^'lU + QtG. 



0X= 



whole area 



This is evidently 

HI+3.JK+5.LM+7 .NF + etG. 



P c 







A Q D 

Fig. 8, 

Another pair of boundary lines like OG and DF must now be 
chosen and the work repeated before we can find the actual position 
of the centre. 

Ex. 2. The Imlf ordinates of the load water-plane of a vessel 
are 12 feet apart, and their lengths are O'S, 3-8, 7-7, US, 14-6, 16-6, 
17-8, 18-3, 18-5, 18-4, 18*2, 17-9, 17-2, 15-9, 13-4, 9-2, and O'S 
feet respectively. 

Calculate {a) the total area of the plane. Ans. 5280 sq. ft. 

{h) The longitudinal position of its centre of gravity. 

Ans. 102 feet from P* ordinate (0-5). 

(c) The displacement in cubic feet per inch immersion at this 
water-plane (this is area in square feet-f 12). Atis. 440 cub. ft. 

The moments of inertia of the area about its centre line and about 
a line through its centre of gravity at right angles to its centre line 
are of importance in calculating the stability of a vessel ; to find 
them we have easy exercises of much the same kind, but perhaps I 
had better not give them here. See my Applied Mechanics, p. 138. 

Ex. 3. There is a homogeneous body symmetrical about an axis; 
the following are its areas of cross-section A at the distances x inches 



MENSURATION EXERCISES 



95 



from one end. Find its volume and its centre of gravity. As it is 
homogeneous, I shall take volume to mean mass in the calculation. 
The whole length is 200 inches. 



X 


10 


30 


50 


70 


90 


110 


130 


150 


170 


190 


A 


320 


304 


311 


297 


292 279 


287 


274 


263 


251 



Ans. The sum of the values of A is 2878, so that the average 
section is 287 -8 square inches. This multiplied by the whole 
length is 287*8 x 200 = 57,560 cubic inches. 
To find the centre of gravity. We imagine slabs measuring 
20 inches axially for each given section. Find, then, the sum 
(20 X 320 X 10) + (20 x 304 x 30) + (20 x 31 1 x 50) + etc. 
This is evidently 

200{(320 X 1) + (304 x 3) + (311 x 5) + etc.} or 5,528,800. 
This divided by the whole volume is 96 inches, the, value of x of 
the centre of gravity. 

55. Ex. 1. An irregular area has the following breadths measured 
at right angles to the direction 00, which we choose, at random, to 
call the direction of its length. 

Two lines OA and O'M, 3*62 inches apart, are at right angles 
to 00' and touch the figure at its ends A and M. I measure the 
breadths of the figure at right angles to 00' at the distance o: from OA. 



Breadths in 
inches 





•75 


■■" 


1-62 


1-73 


1-71 


1-78 1-95 

1 


1-82 


1-47 


1 
0-951 


Corresponding values 
of X in inches 





•123 


0-426 


0-823 


1-22 


1-72 


2-34 


2-57 


2-97 


3-25 


3-47 3-62 



Suppose these numbers to be given, but no other information. 
Of course, if the figure itself were given, we should not need to 
proceed in the following way : 

Plot the breadths and values of x on squared paper and draw 
a curve. 

This curve will enable us to get other breadths. For example, 
we can get breadths for equal increments of x and use Simpson's 
rule. Or we may find the area by the planimeter, and this is the 
area of the original figure. Or we may take the mid-ordinate rule. 

Thus, on dividing the length into 10 equal parts, and measuring 
ordinates at the middles of each, I find the following : 



X 


0181 


0-543 


0-905 


1-267 


1-629 


1-991 


2-353 


2-715 


3-077 


3-439 


Breadth 


0-96 


1-53 


1-65 


1-74 1-71 1-72 


1-80 1-97 


1-70 


1-05 



96 ELEMENTARY PRACTICAL MATHEMATICS 

Now the sum of these is 15*83, and dividing by 10 we have the 
average breadth of 16-08. The length being 3 62 inches, the area 
is 1-583 X 3-62 = 5-73 square inches. 

By the rule of Art. 54 we find the distance of the centre of area 
from OA to be 1 '85 inches. 

Ex. 2. The following are the areas of cross-section of a body at 
right angles to its straight axis : 



A or area of cross-section 
in square inches 





75 


145 


162 


173 


171 


178 


195 


182 147 


95 





X inches the distance of the ^ ; ,0.0 
section from one end | 


42-6 82-3 122 


172 


234 


1 1 
257 297 ' 325 

i 


347 


362 



Plot A and x on squared paper. The average value of A is easily 
found to be 158*3. This multiplied by the whole length 362 is 
362x158-3 = 5-73x10^ cubic inches, the volume. The average 
value of A is, of course, found in the same way as in the last case. 

The centre of gravity of the body is found to be 185 inches from 
the end from which x is measured. 

Ex. 3. Find the volume of a reservoir when its greatest depth is 
42 feet, given the following areas, A square yards of the surface of 
water when this surface is h feet vertically above the lowest point 
of the bottom : 



A 





2100 


8200 


13,100 


15,500 19,500 


25,400 


32,400 


47,100 


52,000 


h 





5 10 


17 


21 


25 


29 


33 


38 


42 



Plotting A and h on squared paper and taking the average height 
of the curve, either by Simpson's rule or the mid-ordinate method, 
or the planimeter, we get the average A to be 20,000 square yards 
or 180,000 square feet. This multiplied by 42 gives 7,560,000 
cubic feet, the answer. 

Questions like the following are often set in " Naval Architecture," 
but the ordinates given are usually equidistant. 

Ex. 4. The areas of successive water-planes of a vessel at the 
following draughts are as follows : 



^=Area(sq. ft.) 


14,850 


14,400 


13,780 13,150 


11,570 


9,200 


6,400 


h = Draught in feet - 


23-6 


20-35 


171 


14-6 


10-1 


5-6 


2-6 



(1) Find the volume displaced and also the displacement of fresh 
water in tons, when the draught is 23-6 feet. 

Ans. 265,000 cubic feet; 7370 tons. 



MENSURATION EXERCISES 



97 



(2) Draw a curve showing displacement V in cubic feet (or T tons 
if in fresh water) and draught. 

(3) If the water-plane area is A square feet, {a) What is the 
extra tonnage displaced in fresh water when the draught is 1 inch 
more % (b) What is the extra displacement in cubic feet 1 

Ans. (a) ^-=-432, {h)A^U. 

(4) Draw curves on one sheet showing for all values of h the 
values oi F, A, T in fresh, and F, A, T^ in salt water. 

Ex. 5. One man on the front of a tramcar looks at a well- 
damped spring balance inserted between the draw-bar and the car, 
thus measuring the pull on the car in pounds. I call this pull F. 
Another man has a means of measuring x, the number of feet passed 
through from some mark on the line. A third man has a watch 
and notes the time t seconds that have elapsed from some arbitrary 
time. They make simultaneous observations of F, x, and /, which 
are given in this table : 



F 


650 


630 


615 


585 


540 


510 


460 


450 


450 


500 


550 


X 


500 


600 


750 


870 


950 


1100 


1300 


1400 


1500 


1650 


1800 


' 





10 


21-71 


29-37 


34-02 


42-46 


53-44 


59 03 


64-78 


73-69 


82-5 



1. Plot F and x on one sheet of paper, and find the average 
value of F. 

2. Plot F and t on another sheet of paper, and find the average 
value of F. 

It is worth your while to think of the reason why these averages 
are not equal. 

As an exercise on the work of Chapter XVIII. ; if x and t are 
plotted as the co-ordinates of points on squared paper, the slope of 
the curve gives the speed v of the car in feet per second ; let this 
Fv 



be tabulated. 



-—- is the horse-power ; let this be tabulated. 
ooO 



56. Ex. 1. Give on one sheet curves which show roughly how 
the number of radians and the sine, cosine, and tangent of an angle 
alter as the angle alters from 0° to 90°. Fig. 9 shows the result. 

Thus OC represents 90°, OB represents 50°, BC shows to scale 
0-6428 the cosine of 50°, ^aS' is the sine of 50° or 0-7660, BPi is the 
radians of 50° or 0'8727 ; BT is the tangent of 50° or M918. 
OR' is straight. OSS' droops downwards and is horizontal at S'. 
OTT' goes up to infinity. CCC is exactly like OSS' as seen in a 
looking glass. 

When a student knows how to find the sine,' etc., of angles, not 
merely between 0° and 90°, but of all sizes, he ought to plot the 
P.M. G 



98 ELEMENTARY PRACTICAL MATHEMATICS 



above curves, say from - 360° to + 360° as an exceedingly interest- 
ing exercise. 

U5r 




O 10 20 30 40 50 60 70 80 90 
Fig. 9. 

Ex. 2. The average height of the positive part of a sine or cosine 
curve is the fraction - or 0*6366 of the greatest height or amplitude. 

Test this. I have only to add sin 5°, sin 15°, etc., sin 85°. I get 
5-7369, and, dividing by 9, I find the average to be 0-6374. To 
get the answer more accurately, I may add sin 2° -5, sin 7° "5, etc., 
and find the average. 



CHAPTER XII. 



SQUARED PAPER. 



57. When one quantity, say y, is expressed as an algebraic 
function of ^another, which I shall call x, take any value of oi and 
calculate the corresponding y; plot on squared paper; draw the 
curve which passes through many such plotted points. 

One most important function to plot is 

y = ax'\ where a and n are any numbers whatsoever. 

Thus y = 9x, or y = 2x-, or y = S'ox^, or y= lOa;^, 

or y = bx^, or etc., 

or y = 9«~\ which may be written xy = 9, 

or y = 9x~\ or y^ 4:X~'\ or etc., 



or 



2x- 



ory = 2x'-^, etc. 



Perhaps it might be well to use 1 as the value of a in every case, 
so that students in a class may draw the whole family of curves 
producible by using different values of n. It is good to prick them 
through on one sheet of paper. 

As an example I will take, say, 



= x' 



Choosing a:= -1, I find «/ = (-l)-°-^^= 1-762, 
„ x=% „ 2/=(-2)-»-^=l-486, 
and, in fact, by choosing the following values of x, I get the follow 
ing values of ^ : 



•2 



10 



11 



y 1-762 1-486 1-345 1-253 1-186 1-133 1-091 1056 1-027 1-000 0-977 



Having plotted these points and drawn the curve, find the 
average value of y between a; ^^ 0*3 and x=}'l. 



100 ELEMENTARY PRACTICAL MATHEMATICS 

Atis. Measuring the average breadth in the usual way, I find 

that the average y is Til 4. 
The area then of the curve between a; = 0-3 and .7;= 1-1 is 

M14x0-8 = 0-891. 
Ex. 1. In y = b + ax% (1) 

we see that J is a mere constant addition ; a merely determines the 
scale of measurement ; hence, if we study 

2/ = ^", (2) 

we may be said to have a complete study of (1). 

The values of y for many values of x have already been computed 
in an exercise of Chap. II. for the following values of «, 4, 1, i, 
- -J , - 1 , - 4 ; so these six curves may be plotted at once. 

Let a number of students get together; let each of them take 
one curve and plot it. When all are finished let them be pricked 
through upon one sheet of paper. Curves with various values taken 
for n are shown upon Fig. 10. They may be said to form one 




Fig. 10. 



family. I give a separate set of drawings (Fig. 11) for the negative 
values of n. 



Ex. 2. Study the family of curves represented by 

y = ae'"' 



(1) 



SQUARED PAPER 



101 




Fig. 12. 



102 ELEMENTARY PRACTICAL MATHEMATICS 



As before, a may be taken as 1, and the family of curves 

y-e"^ (2) 

may be drawn upon one sheet by students, each of whom takes a 
particular value of b. The work is easy if one has a table of 
Napierian logarithms, because 

logjj = bx (3) 

Or if one has only a table of common logarithms, 

\og,,y = 0'^3mx (4) 

Curves with various values taken for b are shown (Fig. 12). 
Values of y for various values of x have already been computed in 
Ex. 4, Art. 28, for the following values of ^, 4, 1, J, -J, - 1, - 4; 
so these six curves may be plotted at once. 

58. The Position of a Point. — The following story is not true. 
During the Seven Years' War, in 1760, in Saxony, a gentleman 
buried treasure in four different places on his estate. He was 
suddenly killed. His son did not know where the treasure was 
buried ; he knew that it was buried. But there was a parchment 
document which his father had confided to him containing these 
symbols, ^ ^ 2000, y = 977; x= - 560, y = 700 ; 

x= -750, y= .-650; x = 35Q, y= -274. 
He sought for the treasure in vain. 

In 1860, a descendant, a young American, found his way to the 
old estate, and was welcomed as a distant cousin. He was not very 

rich, yet he fell in love with a 
girl cousin as poor as himself. He 
had long known the old legend 
about the buried treasure — indeed, 
no member of the family ever forgot 
it. He happened to pick up a 
- school-book one day and saw in 
it this figure (Fig. 13), which he 
had never seen before, albeit it 
is very common in mathematical 
books. It is so very common 
that no German who has gone to 
school from the age of seven to 
the age of 25, and had himself 
stuffed like a Strasburg goose with mind-training knowledge nine 
hours a day all that time, would ever dream of connecting it in any 




Y 

Fig. 18. 



SQUARED PAPER 



103 



way with legend or romance, or, indeed, with anything but a school- 
book. But to this young American it told a story. Here were x's 
and y's evidently distances from two well-marked straight lines. 
So he and his cousin looked up an old 1760 map of the estate, and 
sure enough they saw two faint 
lines drawn in continuation of Y 

the sides of the very same old 
building in which they then 
were, something like Fig. 14. 
Then George W. went to Doro- 
thea's father and said, "If I 

discover the old hidden treasure 

will you let me marry my 
cousin r' "Here are two dis- 
coveries," said old Heinrich, 
"one by you and one by me. 
I know well that an American, 
who has no such school-book 
knowledge as our Germans are 

hampered with, has imagination and ability to do things, and he is 
not dull ; and therefore I will let you marry Dorothea, even if you 
have made a mistake in this thing." Then did George Washington 
Ollendorff make this sketch (Fig. 15) for his uncle. 



Fio. 14 



700 

b" :b' 



O B" 



: X 



6^0 



1274. 

•,p"' 



Y 

Fig. 15. 



"What the distances are in," he said, "feet or yards, I do not 
know ; but you know what measure they used here a hundred 



104 ELEMENTARY PRACTICAL MATHEMATICS 

years ago, and the places I have marked are the spots to dig for 
buried treasure." 

Then did old Count Heinrich von Ollendorft' turn wrathfully upon 
the guileless George W. and say : 

" Now do I see that you have been deceiving me all this time, 
for you know about Cartesian Co-ordinate Two-dimensional Analytical 
Geometry, and although nobody agrees with me on this matter, I 
affirm that you must be dull and stupid." 

But Dorothea placated her father, and said, " You know, dear 
father, that I know nothing of learning, and yet even I see that if 
AP is 2000, being measured to the right of a line, A'P' may well 
be called - 560, being measured to the left of the same line. Also, 
if FB is 977, being measured above the line, P"'B"' may well be called 
- 274, being measured below the line." 

Also, George W. spoke up and said, " I guess I know precious 
little of that terrible science you mentioned just now, but I say 
that this is a common-sense way of bringing together the legendary 
figures and the legendary map. Besides, I have your promise, 
and I here declare that for myself I will take none of this treasure ; 
my treasure was buried here at the place of our origin." 

But old Heinrich insisted on dividing the four buried treasures 
among the 207 Von Ollendorffs of Europe and the 310 OllendorfFs 
of America, and the joint shares of George W. and Dorothea 
amounted to 523-07 dollars. 

59. When a point is not on a fiat surface, we must still give two 
dimensions to find its position. Latitude and longitude are quite 
familiar to you all; thus, for example, if I say that a small island 
has been discovered in 56° N. latitude and 30° W. longitude, it is 
easy to find its place on a map or a globe. A shipmaster can tell 
his position with a possible error of only a mile or so, if his 
chronometer is to be depended upon, and if he can see the sun. 

Ex. 1. Plot the curve ^ + ^ = 1. 

It is easy to see that this is the same as 



= l-% or y=±hJ\A 



There are then two values of y for each value of x^ and again it 
is evident that ic=+4or«=-4 will produce values of y that are 



SQUARED PAPER 105 

equal to one another. To make this clear one must work a few 
exercises. 

(1) Suppose 6 = 5 and a = 5, the formula simplifies to 

y= ±v/25^^ 

Taking x=l, then i/= + v/24 and also y= ~ sl'2i ; that is, there are 
two values of y iov x=\, and so we get two points. Indeed we get 
four points by the one calculation, for x= -\ would give the same 
answers for y. 

Now take x= ±2 and calculate y, and so get other four points, 
and so on. The curve is an ellipse whose diameters are equal to 
one another, that is, a circle. 

(2) Suppose a = 5, 6 = 3. 

Then y= ±l'j2b-xK 

If you have done (1) you see that this will be very easy. 

Ex. 2. Plot the hyperbola 

«2 62 ^• 
This is evidently the same as 

y=±-s/xJ-a\ 

Take h-=Z and a = 5, and so plot the curve 
y^ ±|x/^2i:^5. 

Ex. 3. The Cycloid. Instead of giving y as a function of x or 
giving an equation connecting x and y, it is more convenient to 
state that 3. ^ ^ ^(^ _ gjjj <^^ ^jj^j y = a{\- cos <^). 

Take a value for ^, calculate x and y. Remember that <^ is in 
radians. 

Take another value for 0, calculate a new pair of values for x and y. 

In this way get the co-ordinates x and y and plot points of the 
curve ; ^ is an auxiliary angle. 

These calculations were performed in Ex. 17, Art. 28, it only 
remains now to plot the curve. 



CHAPTER XIII. 



THE LINEAR LAW. 



60. When a curve is simple looking, it may often be expressed 
by a simple algebraic formula. 

When we plot corresponding values of y and x, and find that the 
points lie in a straight line, we always find that there is a simple 
law connecting them of the form 

y = a-\-bx, 
where a and b are constants. 

It is worth while spending some time in plotting this function. 

Thus, plot y = 2 + 0'75x. By taking the following values of ic, I 
have calculated the corresponding values of y. Let these be plotted 
on squared paper. You will find that the points lie exactly in a 
straight line. 



X 





1 


2 


3 


4 


5 


6 


7 


8 


y 


2 


2-75 


3-5 


4-25 


5-0 


5-75 


6-5 


7-25 


8 



I find that a stretched black thread gives the best test of 
straightness. 

Now try «/ = 2 + 0-5a:, ^ = 2 + 0-9a;, 

i/ = 2-0-3a:, y = 2-0-75ic. 

In every case you will find a straight line. 

Now notice that in all these cases we have the same value of a, 
and consequently all your lines have some one thing in common. 
What is it % Find out for yourself. 

61. Now plot another series of straight lines y = a + bx, in which 
b is common to all. 

Take y=l+0'75x, y = S + 0-1bx, y = 4: + 0'75x, 

«/ = + 0-75«, ^ = - 1 + 0-75x', y= -2 + O^S.^. 



THE LINEAR LAW 



107 



You will find all such lines with the same b have the same slope, 
and indeed I usually call b the slope of the line. These lines are 
all parallel to one another. 



62. Slope of a Line. 

corresponding y ; it is 



If y = a + bx, take x = x^, and find the 



Now take a new Xy call it a;.^, and find the corresponding y ; it 
is ?/2 = (i + bx2. Subtracting, we get 



or 



or in words. 



increase in y 



b. 



increase m x 

Hence whatever values we may tak« for x^ and x^j we find : 
The increase of y divided by the increase of x is a constant, b. 
Now it is the rate of increase of one thing relatively to another 
which enters into most of our thinking about things, and we notice 
that this rate is constant in any case when on plotting the quantities 
on squared paper we get a straight line. 

63. In the laboratory, when we have measured corresponding 
things, and on plotting we find the points lying in a straight line, 
we are usually glad, because we know that the things are connected 
by a very simple law. Besides, it is a law which it is very easy to 
test with a black thread. 

Ex. 1. The following observed numbers are known to follow 
a law like y = a-\-bx, 

but there are errors of observation. Find by the use of squared 
paper the most probable values of a and b. 



X 


^ 


3 


4-5 


6 


7 


9 


12 


13 


y 


5-6 


6-85 


9-27 


11-65 


12-75 


16-32 


20-25 


22.33 



On plotting, as in Fig. 16, and stretching a black thread to get 
the straight line which lies most evenly among the points, I find 
that two points in my selected straight line are 

a; = 2, ?/ = 5-5 ; and X = 10, ?/ = 17-5. 

Hence, to make y = a-\-bxf\.t these numbers, 

5-5 = a + 2^^, 17-5 = a+ 106. 



108 ELEMENTARY PRACTICAL MATHEMATICS 

Subtracting, we have 

12 = Sb or 6 = 1-5; 
therefore 5*5 = a + 2 x 1 -5, 

or a = 2'5. 

Hence the law required is 



22 




















































y 














































^ 
















































A 




















































/ 




















































/ 


















































,/ 


y 


















































Y 
















15 




































/ 














































■ 


/ 


/ 


















































/ 




















































A 


• 
















































/ 


•/ 




























10 






















/\ 


















































/ 


















































/ 


/' 


















































/^ 




















































/^ 
















y 


= 


2' 


5H 


VI 


5 


X 














5 








/ 


^ 
















































y 




















































Z' 




















































/ 


























































































































































u 







5 6 7 

Fig. 16. 



8 9 10 11 



12 13 



Nott. — The M'ooden -headed, cock-sure academic person will tell you that he 
has an algebraic method of infinite exactness based on the laws of probability 
for finding the best values of a and h. Do not believe him ; the black-thread 
method is easy to understand, and one therefore has one's wits about one when 
using it. To the average man using the other method it is occult ; his belief 
in it is like our ancestors' belief in magic. Even the good mathematician 
forgets that it is based on the assumption that every observation is as likely 
to be in error as every other one. But with the black thread one cannot help 
adopting rules as to probability which suit the nature of the observations, 
especially if one has made them oneself. Very often the probable errors are 
not all equal, but rather the percentage error or the probable error may be in 
some curious relation to the observations. 

Besides, there may be an absurdly large error in one of the observations. 
Our method shows a possible large error at once, and suggests a repetition of 
the experiment. 

The student ought now to calculate y for each of the above values 
of x^ and so see the probable errors of the observed values of y. 



THE LINEAR LAW 



109 



Ex. 2. In the price list of non-condeiising steam turbines driving 
electric generators, I find the following : 



Maximum output of plant 
or power in kilowatts = K. 



Price in pounds sterling 
= P. 



Speed in revolutions 
per minute =«. 



2,000 

1,000 

500 

100 



15,785 
8,085 
4,235 
1,155 



1,200 
1,800 
2,500 
3,500 



Show that if we plot K and P we get a straight line. Now, in 
many motors it is K-^n, which, when plotted with P, gives a 
straight line ; show that this is not so here. What is the list price 
of turbine plant of 700 kilowatts 1 Ans. P= 385 4- I'lK; £5775. 

Ex. 3. The following tests were made of a steam turbine 
(condensing) electric generator : 



Output in kilowatts. 
K. 


Steam consumption 

per hour in lb. 

W. 


1,190 
995 
745 
498 
247 



23,120 
20,040 
16,630 
12,560 
8,320 
4,065 



The steam was not all in the same state, being superheated from 
10 to 20 degrees Centigrade. Find the best straight line law. 

Ans. JF=l6-2K+4220. 

Ex. 4, I made the following tests of a large single cylinder gas 
engine. / is the indicated power and B the actual or brake horse- 
power given out by the engine. Plot / and B, and find the law 
connecting them. I did not measure B when /is 100; if I had 
made this measurement, what, in all probability, would my answer 
have been 1 



Brake horse-power B - 


16 


57 


95 


99 


117 


Indicated horse-power / 


35 


73 


114 


120 


139 



Ans. ^ = 0-967- 16-4. 



When 



7=100, ^ = 79-6. 



Ex. 5. If P is the electric power in kilowatts sent out of an 
electric lighting station, and if C lb. is the amount of coal burnt in 



no ELEMENTARY PRACTICAL MATHEMATICS 



the boiler furnaces per hour, find the law connecting P and C if the 

following tests are correct : 

Ans. O=l-67P + 540. 
It is interesting for the student to calculate C-^P 
in each of these cases, noting the diminution of effi- 
ciency of an electric light station when giving out 
small supplies of power, that is, during the greater 
part of the 24 hours. It is worth while to draw a 
curve showing the value of C-rP for each value of P. 
Of course C-^P means pounds of coal per kilowatt 
hour. 

Ex. 6. When the weight A was being lifted by a laboratory 
crane, the handle effort B (the force applied at right angles to the 
handle) was measured and found to have the following values : 



p 


c 


349 


1121 


291 


1020 


228 


927 


171 


820 


119 


743 


71 


652 



A 





50 


100 150 


200 


250 


300 


350 


400 


B 


6-2 


7-4 


8-3 


9-5 


10-3 


11-6 


12-4 


13-6 


14-5 



What law connects A and B 1 Ans. B = 0-0207^ + 6-3. 

Ex. 7. The speed ratio of the above crane being 80, a handle 
efi'ort ^ -^ 80 ought in every case to lift A if there were no friction 
or other source of unnecessary work done. Dividing ^ -=- 80 by B, 
that is finding A -^ SOB and calling this the efficiency e, plot e and A, 
and so find a curve showing how the efl&ciency of the crane depends 
upon the load. 

Ex. 8. At a certain electricity works, if 7V is the annual works 
cost in millions of pence and 2' is the annual total cost and U the 
number of millions of electrical units sold, it is found that approxi- 
mately since 1894 

JF= 0-3 + 0-6 U, T= 0-5 + 0-97 U. 

If, in 1901, U was 3, and if U increases by 0*5 each year, what 
will be the value of U m 1904, and what will be the probable cost, 
and also the cost per unit? [This question was set in 1902.] 

64. There are many operations of which the following is an 
example : 

An examiner has given marks to papers ; the highest number of 
marks is 185, the lowest 42. He desires to change all his marks 
according to a linear law, converting the highest number of marks 
into 250 and the lowest into 100. How may he do this, and what 
is the converted number of marks of a paper, the original number 
being 140"? 

(1) Laborious Metlwd. For any paper let y be the converted 
number of marks and x the original number. 



THE LINEAR LAW 

The assumption is that ,^ = « + hx, where a and h are constants 



111 



Then 



Subtracting, 



250 = a+ 1856 
100 = «+ 426 



150= 1436 
6=1-048, 
100 = ^ + 42x1-048, sothatft = 56. 
Hence he will convert all his marks by the rule 

3/ = 56 + l-048rr. 
Thus, to convert x= 140, 

he has y= 56 + 1-048 x 140 

or y=203. 

(2) Using Squared Paper. Plot the two points a; = 185, «/ = 250 and 
ic = 42, y = 100 ; join these points by a straight line and read off the 
y corresponding to any x. 

(3) Let him have two scales, one of them marked on a uniform 
strip of vulcanized india-rubber. Let him stretch the india-rubber 
scale alongside the other, making the marks 185 and 250 agree, 
and the marks 42 and 100 also. Fastening the scales together, he 
easily reads off the converted number corresponding to any original 
number. He is here depending upon the uniformity of stretch of 
the india-rubber. 

(4) Best MetJwd. Using squared paper and a handy scale. Let 
the scale lie sloping on the squared paper so that the divisions 
correspond. 

There are other quite different ways of proceeding. 



EXERCISES. 

1. On testing the relation between the weight W lifted by the pulley 
blocks in the laboratory and the effort E required to lift it, the following 
numbers were obtained : 



w 


' 


14 


21 


28 


35 


42 


49 


56 


E 


3-0 1 51 


7-4 


9-6 


11-5 


14-0 


160 


18-25 



Try if they are connected by a law of the form E=aW+h, and if so, 
find the best values of a and 6. Am. ^=0-303 W+O'M. 

2. Try whether the following experimental values are connected by 
the linear law, and if so, find the best constants. 



X 


14 


28 


42 


56 


70 


84 


98 . 


112 


y 


3-5 


5 


6-75 


8-25 


9-75 


11-5 


13-25 


14-78 



Arts. v = 0-116^+l-75. 



112 ELEMENTARY PRACTICAL MATHEMATICS 

3. The following numbers are from a student's laboratory note-book : 



R 


9-35 


9-99 


10-41 


10-70 


10-77 


11-186 


11-50 


t 


13 


27 


40 


50 


51 


60 


70 



R and t are supposed to be connected by a law like R=R(^{\-\-at). 
Try if this is so, approximately ; and if so, what are the best values of 
^nanda? Ans. /2o = 8-854, a=0-00431. 

4. An electrically driven pump, being tested, gave the following results. 
E is the electrical horse-power given to the pump, H is the horse-power 
actually spent in raising the water. 



E 


3-12 


4-5 


7-5 


10-75 


H 


1-19 


2-21 


4-26 


6-44 



What is the law connecting E and H'l The efficiency of the plant 
being e= H-^E, express it in terms of E. 

Ans. ^=1-45^+1-38, e=^-^(l•45^ + l-38). 

5, K kilowatts being the average electrical power actually delivered to 
customers from an electric station during the 24 hours, W the average 
weight of coal consumed per hour, the following observations were made : 



K 


2560 I 2100 

1 


1800 


1520 


1300 


W 


7760 


6740 


6110 


5480 


5030 



Find the law connecting K and W. 

The maximum power which might be delivered to customers is 13,060 ; 

W 
let A713,060 be called /, the load factor ; let — be called iv, the coal per 

unit. [The Board of Trade 'unit' of energy is one kilowatt hour.] 
What seems to be the law connecting w and/? Calculate w when /"has 
the values 0-25, 0*20, 0-15, 0-10, 0-05, and tabulate w and/. 



•1689 



/ 


0-25 


0-20 


015 


010 


0-05 


w 


2-843 


3-012 


3-293 


3-856 


5-545 



Ans. «; =2-167 4- 



6. If ?/ = 20 + \/30 + .r2. Take various values of x from 10 to 50, and 
calculate y. Plot on squared paper. What straight line agrees with the 
curve most nearly between these values? Ans. y = 0-97.^ + 21*5. 

7. An engineer wanted to be able readily to state approximately the 
total cost K of steam plant, that is of the buildings, boilers, pipes, fittings, 
foundations, and engines for any given maximum indicated horse-power /. 
He obtained actual figures from a number of people who had recently 



THE LINEAR LAW 



113 



(1896) put up good steam plant under the average conditions existing in 
and about English manufacturing towns. Taking rough averages, he 
found : 

If /= 200, K may be as great as £4600 or as little as £3800 ; the average 
may be taken to be £4200. 

If /= 30, K may be as great as £900 or as little as £550 ; the average 
may be taken to be £725. 

If /= 120, K may be as great as £2600 or as little as £2300 ; the average 
may be taken to be £2450. 

Plotting on squared paper, I find that a straight line will lie fairly well 
among the points, and this leads to the following rule : 

^=100 + 207. 

What is the probable cost of steam plant whose indicated power is 160? 

Ans. £3300. 

8. The following figures have just been published (May, 1912) ; tests 
of a Diesel oil engine using heavy oil. W is the weight of oil per hour, 
B is the brake horse-power. Full load means 250 brake horse-power. 
w is T^-^ B^ or weight of oil per hour per brake horse-power. 





\ load. 


\ load. 


1 load. 


Full 
load. 


Ten per cent, 
overload. 


xo 


0-64 


0-50 


0-45 


0-448 


0-458 



From these figures I have computed the values of W and B 



B 


275 


250 


187-5 


125 


62-5 


W 


126 


112 


84-4 62-5 


40-0 



Plotting on squared paper, I find a straight line, except for the over- 
load case. Neglecting this case, which is evidently outside the simple 
law, I find lF=17-6-fO-375, 



and therefore 



..=-g=-g-fO-37. 



65. Exercise. The keeper of a restaurant can entertain at 
most 1 20 guests of the average kind in one hour ; take it then that 
during the day he might have 1440 guests. After some time he 



had observed these average results 



Daily number 
of griests Cr. 


Expenses, rent, rates, 

taxes, wages, and 
maintenance E pounds. 


Cost of food 
and drink 
F pounds. 


Total money 
collected 
M pounds. 


Profit 
P pounds. 


240 
360 


8 
9 


10 
14-7 


18 

27 



3-3 



(a) Taking linear laws, show that 

P = 0-0275G^-6-6, M = 0'01bG, i^=0-039(?-H0-6, E = Q-^^\^G. 
P.M. H 



114 ELEMENTARY PRACTICAL MATHEMATICS 



(b) What number of guests will just produce a profit of £4 per dayl 

Ans. 386. 

(c) What is his profit per guest in the three cases when he has 
300, 600, or 900 guests 1 Ans. 1-32, 3-96, 4-85 pence. 

(d) It will be noticed that the average guest is supposed to pay 
18 pence ; suppose the above linear laws for E and F and G were to 
hold, whether the guests crowd in at particular hours or distribute 
themselves more uniformly during the day ; suppose the guests at 
various hours are as follows, and in the first case that every guest 
pays 18 pence at whatever hour he comes, and in the second case he 
pays less according to the scale specified. Compare the daily profits : 



Time of day. 


o 


1 


S 


3 


c^ 


CO 






t 




t 


•* 


1 


1 




Guests in one 
hour 


2 


3 


10 


90 


120 


30 


3 


2 


5 


50 


80 


10 


2 


2 


o 


Pence paid by 
each guest 


18 


18 


18 


18 


18 


18 


18 


18 


18 


18 


18 


18 


18 


18 




Guests in one 
hour 


40 


60 


80 


80 


80 


80 


80 


80 


80 


80 


80 


80 


80 


80 




Pence paid by 
each guest 


12 


12 


14 


17 


18 


17 


16 


14 


16 


17 


18 


17 


16 


16 



The daily guests in the two cases are 409 and 1060 respectively, 
and we can calculate F and E from our formulae. 
Hence we have the results : 



Daily number 
of guests G. 


Expenses 
E pounds. 


Cost of food, etc. , Money collected. 
F pounds. Pounds. 


Daily profit. 
Pounds. 


409 
1,060 


9-41 
14-83 


15-73 
39-8 


30-7 
70-33 


5-56 
15-70 



It is on this system of charging less when business is slack 
that lodging house and hotel keepers proceed ; the Electric Light 
Companies soon discovered that the system was a good one, charging 
less per unit during the slack hours of the day. The above example 
shows that it might be important for a restaurant keeper or, indeed, 
almost any person in any kind of business to follow the same 
method. 

If the student will use the name "load factor/" for 6^-^1440, he 
will be using the language of the supply engineer. 



CHAPTER XIV. 



SQUARED PAPER. 

66. Laws convertible into Linear Laws. Of course, if we have 
some theory to guide us, it is easy to find the algebraic law connect- 
ing X and y. The following are observed quantities : 



X 


11 


1-8 


2-5 


2-9 


3-6 


4-3 


4-8 


5-4 


y 


1-91 


213 


2-42 


2-65 


309 


3-66 


4 09 4-73 



Plotting on squared paper, we find a regular, simple curve. For 
many purposes this curve is enough ; it allows us to correct observa- 
tions, to interpolate, etc. But if we suspect that there is some simple 
algebraic law connecting y and x, we trust to our experience of curves 
and to lucky guessing and trial to determine the actual algebraic law. 

After trying various things, let us suppose that we are lucky 
enough to think of plotting y and x^, or let us suppose that theory 
tells us to plot y and x^. Squaring all the values of x, we get : 



a^ 


1-21 


3-24 


6-25 


8-41 


12-96 


18-49 


23-04 


29-16 


y 


1-91 


213 


2-42 


2-65 


3-09 


3-66 


4-09 


4-73 



I hnd on plotting now that I get a straight line. As there are 
evidently errors in the given values of x and y, I stretch my black 
thread and settle what is the best possible straight line : that is, 
I fix some two points as being probably quite correct. In this 



case I fix 


2/ = l-9 when a;'^=l, 




y = 4-3 „ ^2^25 


Hence, if 


y = a-\-hx'^^ 




l-9 = ft + ft 




4-3 = a + 256. 



116 ELEMENTARY PRACTICAL MATHEMATICS 



Subtract 2-4 = 246, or b = Ol, 

l-9 = « + 0-l, or rt = l-8. 

Hence y=VS + 0'lx'^ 

is the law. Now taking the above values of x and calculating i/, 
I can find the probable errors in the observed values. 

67. Very often an experienced man who knows a great deal 
about curves fails to discover a simple law, although it may exist. 
If the points do not lie approximately in a straight line, I often try 
y = ax^ or y = ae^''; but it depends upon the appearance of the plotted 
curve what I shall try. 

I remember the first time I discovered a law all by myself. It 
was in 1875.* I was a very inexperienced young man, and thought 
I had discovered a law of as much importance as Newton's law of 
gravitation. Of course I now know that many other empirical 
formulae would have suited my numbers equally well ; the merit of 
a formula lies not merely in its fitting the numbers, but also in its 
simplicity. Any single-valued function whatsoever may be repre- 
sented by a formula like 

y = a + hx + cx^ + dx^-\-ex!^ + QtG., 
if we only take enough terms, f 

The merit of Newton's law lies in this, that, although it is so 
extremely simple, it seems to be wonderfully true throughout such 
space as we have had a chance of applying it to, the space occupied 
by our solar system. Beyond our system we know nothing about 

* See Proc. Royal Society, 1874-5. 

I had observed carefully and found the following results for the con- 
ductivity C of glass at varying temperatures 6 : 



Temp. Fah. 6 


58° 


86° 148° 


166° 


188° 


202° 


210° 


C 





0004 


0-018 


0-029 


0-051 


0-073 


0-090 



I plotted 6 and G on squared paper, and tried in all sorts of ways to find 
the algebraic law connecting thera, for it was evident to me that some simple 
law did connect them. At length I was lucky enough to think of plotting 
log C with 6, and found that my points lay in a straight line, and it was then 
easy to show that (7=0-000124 x 1-032^. 

t At the Royal Society, once, I heard a mathematical man talking about 
an empirical formula he had discovered. It was a good enough formula, 
within the range of values of his experiments ; but he actually discussed the 
possibility of unreal values and curious roots of equations altogether out of 
the region of his experiments, as if he had found an infinitely exact law of 
nature. The exercise of a little common sense will prevent this kind of 
' ' extrapolation. " 



SQUARED PAPER 



117 



laws of gravitation, but in our usual optimistic Avay we always 
assume it to be true, just as we reasonably assume a great many 
other things to be true which may yet be proved to be false. In- 
deed, we do not know that the law of gravitation, as usually stated, 
is true inside our solar system ; the weight of a body may really be 
dependent upon its temperature, for all we know, or some other 
property that we now assume it to be independent of. 

It is astonishing how we go on believing things without proof, 
and so long as we are aware of the fact, there is no harm in it. 
Hundreds of thousands of lecturers on chemistry demonstrate the 
exact composition of the atmosphere to audiences, and then a man 
of original thought like Lord Rayleigh comes along and really tests 
these statements and finds them wrong. 

Exercise. A pulley being fixed, a cord lapping round it by the 
amount I (a lap of amount \ means one quarter of a turn, or 90" 

or - radians), the slack end being pulled by a force M ; the force N 

on the tight side was found to be just sufficient to maintain a steady 

slipping ; find the law connecting — and I. 



First plot —j. and / ; now plot / and log -p. 



The force M was 2. 
















Lap^ 


i 


4 


1 




u 


u 


1^ 


2 


2i 


2^ 


N 


3-17 


5-06 


7-90 


12-68 19-90 

1 ■ 


32-10 


50-12 


80-3 


125-8 


201-5 



Ans. 



N 
M' 



(See Ex. 1, Art. 28.) 



68. Now let me give you an example of a lucky guess. Here is 
the sort of result obtained in many thousands of cases of experi- 
menting with steam engines. [I use one of the very best sets of 
experimental numbers ever obtained from a small condensing triple 
expansion steam engine, tested under seven steady loads, each last- 



ing three hours.] 
















Indicated horse-power, / - 


36-8 


31-5 


26-3 


21 


15-8 


12-6 


8-4 


Pounds of steam used per 
hour per indicated horse- 
power, ic - 


12-5 


12-9 


131 


13-3 


14-1 


14-5 


16-3 



Plot / and w on squared paper, as in Fig. 17, and you will find 
the sort of result obtained by me and many other people year after 



118 ELEMENTARY PRACTICAL MATHEMATICS 

year. Somehow we could make no use of our results ; there was 
no simple law. But Mr. Willans was lucky enough to think of 
plotting with /, not w, but /F, the whole weight of steam used per 
hour by the engine. Now try. Of course Ixw is what I call fF. 



I 


36-8 


31-5 


26-3 


21 


15-8 


12-6 


8-4 


w 


460 


406-2 


344-5 


279-3 


222-8 182-7 137-0 





























































/ 
































































/ 


■St 






450 
























































/ 


































































/ 


































































/ 
































































y 
















400 


\ 














































/ 






















\ 












































/ 


































































/ 




























V 






































/ 






























\ 


































































16 




^ 
































/ 


r 
































\ 






























/ 








































^ 


























/ 


































300 








\ 
























y 


































:? 


15 








\ 




















/ 








































"5 








s 














J 


r 






































^ 










- 


+ 












^i 










































250 




\j 


s, 






/ 














































14 
















^ 


r 
































































<S 
































































/ 








^ 


■ 










































200 














/ 
















L.^ 
















































<f 




















^ 


H 


"-J 




* 


V 




































/ 
































— 




i 


























/ 
















































— ■ 


— . 


r: 


. 






15-5 




7 


/ 




















































































1 


ndcc 


tied 


Hon 


je 


P 


)wer 


























8 


1 











1 


5 








2 


o 








2 


5 








3 











a 


5 








4 






Fig. 17. 



You see by Fig. 17 that the points lie sufficiently nearly in a 
straight line for us to be able to say that 

;r= 37-5 + 11-57, (1) 

a very simple law. If we use Iw for W and divide by /, we get 

^JJ^ + n-5, (2) 

and therefore if it had struck anybody to plot not w and /, but 
w and ^, a simple law would have been discovered.* 

* Note. — In every case tried by Mr. Willans, he found a linear law con- 
necting W and /. It can be proved that in non -condensing engines it is 
reasonable to expect the linear law to hold, but that it is not true in condensing 
engines unless special precautions are adopted to keep the cylinders dry. 
Also, it is true in steam engines only when -we regulate by letting the steam 
pressure vary ; the cut-off is not supposed to be altered. See Ex. 41 and 42, 
Chap. XXVII. 



SQUARED PAPER 



119 



69. Example. The following values of x and y were observed, 
and it was important to see if some simple algebraic law connected 
them. 



X 


•05 


•1 


•3 


•5 


1-0 


2-0 


1-4: 25 


y 





•136 


•26 


•55 


•78 


•97 


122 


11 1-24 



Plotting on squared paper, you will notice that when x and y are 
small, we may say roughly that 

y ccx, 

but as X gets greater and greater, y seems not to get proportionately 
greater, but rather to be approaching a limiting value. In any case 
of this kind first try if 

_ ax 
y~\Th~x 
is true, that is if y + bxy = ax ; 

y 



or, dividing by x, if 



-\-hy = a. 

y 



This will be tested if we plot ^ and y on squared paper. 

X 

In the present case, I find that the points so plotted do lie very 
nearly in a straight line, such that 



and hence 



X 

y-= 



-2y + 3, 
3a; 



l+2a: 



70. The following values of x and y when plotted on squared 
paper give a curve differing in character from the last, principally 
for the smaller values. 



X 


° 


0-5 


1-0 


1-5 


2-0 


2-5 


3^0 


y 





0^48 


137 


212 


2-58 


2^93 


3-07 



The student ought to plot them, and then see the reasonableness 



This may be written 



_ ax"^ 
^ ~ 1 + sa;^ ' 

+ syx^ — ax^ 



or 



^^^sy = a. 



120 ELEMENTARY PRACTICAL MATHEMATICS 



The kw will therefore be very well tested if we plot y and -^ 

on squared paper. If this is done, we find a straight line to lie 
evenly among the points, and I take 



y= 



l+0-6a;2 



as representing the true law. 

71. Exercise. Consider the following observed numbers 



X 


1-70 


2-24 


2-89 


4-08 


5-63 


6-80 


8-42 


12-4 


16-3 


190 


24-3 


y 


320 


411 


491 


671 


903 


1050 


1270 


1780 


2250 


2520 


3180 



If these are plotted on squared paper we get a curve. The shape 
of this curve would suggest to me, 

1. To plot \ogx and log?/, or 

2. To plot X and log y, or 

3. To plot log X and y, or 

4. Try other tricks. 

Now it will be found in the present case that the first of these 
plans succeeds. Thus I find : 



log a; 


0-2304 


0-3502 


0-4609 


0-6107' 


0-7505 


0-8325 


0-9253 


1-0934 


1-2122 


1-2788 


1-3856 


logy 


2-5051 


2-6138 


2-6911 


2-8267 


2-9557 


3-0212 


3-1038 


3-2504 


3-3522 


3-4014 


8-5024 



and on plotting I find that the points lie very nearly in a straight 
line. In the way already described, I find the most probable line 
to be logy-0-876loga; = 2'299, 

y=199a;-<'-^'«. 



or 



72. Exercise. In some experiments in towing canal boats the 
following observations were made : 



p 

Pull per ton in lbs. 


Speed in 


V 
miles per hour. 


0-70 
1-70 
2 -.35 
3-20 
3-50 


1-68 
2-43 
3-18 
3-60 
4-03 


Show that there is a law of the 
P = 0-3 


form 





SQUARED PAPER 



121 



73. Exercise. The following measurements have been made 
upon a steamer : 



/ 

The indicated horse-power. 


D 
The displacement in tons. 


V 
The speed in knots. 


140 

410 

820 

1,500 


1,748 
1,748 
1,748 
1,748 


7 

10 
13 
16 


442 
351 
294 


2,030 
1,400 
1,000 


10 
10 
10 



Find if there is a law of the form 

Ans. By plotting log/ and logi) for the speed of 10 knots, and 
by plotting log/ and \ogv for the displacement 1748, we 
obtain /_ 0-00513/>°-^»V^. 

74. I may say that plotting log x and log y on squared paper is 
so often successful, especially in gas and steam-engine work, that an 
old pupil of mine (Mr. Human) induced a stationer to manufacture 
sheets of logarithmic paper, so that one might lay out logs; and 
\ogy without using a table. Here is a sheet of such paper. You 
will see that the figure 5 is not at the distance 5 inches from the 
axis, but at a distance proportional to log 5. 

This paper is especially valuable when, instead of the law 

yx-=^C, 
the law that is most nearly true is 

(y^o.){x + li)" = C* 

For, after we have plotted y and x upon this curious sheet and 
we find that no straight line will lie among our points, it is easy to 
study, not the plotted points, but points lying one or two or three 
divisions to the right or above, or to the left or below them. 

Exercise. I have found that the expansion curve of an indicator 
diagram always seems to follow approximately a law : 
pv" = constant. 

The following measurements (it is easy to prove that the scales of 
measurement are really of no consequence if we want to find n) are 
taken from a gas-engine diagram. But it is known that the clearance 

* See Chap. XV. for a method of studying all such curves as this without 
using squared paper. 



122 ELEMENTARY PRACTICAL MATHEMATICS 



was not exactly measured, and hence all the values of v may need 
the addition or subtraction of a constant number. Find the law 
and the correction for clearance. 



p 


39-60 


44-7 


53-8 


73-5 


85-8 


113 2 


135-8 


178-2 


V 


10-61 


9-73 


8-55 7-00 


6-23 


5-18 


4-59 


3-87 



Ans. I find, using logarithmic paper, that 0-6 must be subtracted 
from every value of v as the correction for badly measured 
clearance. In fact, I find 

p{v-0'^Y^-= constant. 

75. The following are tabulated values of p, the pressure in 
pounds per square inch of saturated steam ; u being the number of 
cubic feet per pound. It was found that the law 

pu'' = c, a constant, (1) 

Avas very nearly true, and as a formula of this kind is very easy to 
use in calculations, it is important to find the best values of n and c 
within the following range of values : 



p 


6-86 


14-70 


•28-83 


60-40 


101-9 


163-3 


250-3 


u 


53-92 


26-36 


14-00 


6-992 


4 "28 


2-748 


1-853 



If (1) is true, we have 

log I? + n log u = log c. 
Plotting log j9 and log w, I find that a straight line lies very evenly 
among the points, and, in fact, we may say, 

log;? + 1-0646 log ti = 2-68. 
Hence ^!iio646 = 479. 

76. Ex. 1. Prove that the whole weight or displacement in tons 
of a vessel is the volume V cubic feet of displaced water divided 
by 36 if the water is fresh, or divided by 35 if the water is salt. 

At the following draughts, h feet, a particular vessel has the 
following displacements in tons or in cubic feet found by calculation 
from the drawings : 



Draught h feet 


.5 


12 


9 


6-3 


Length on water line I feet 


300 


280 


270 


265 


Greatest breadth on water line 6 feet 


36 


35 


32-5 


27-3 


Displacement V cubic feet 


73440 


52920 


35640 


20520 


Displacement in sea water T tons 


2098 


1512 


1018 


586 



SQUARED PAPER 123 

Show that they satisfy the laws 

Ex. 2. If for each draught A, the length I and greatest breadth 
of the vessel at the water line are as stated, show that 

Ex. 3. Taking the law F= 1545/^1 ^2 to be true, since A the 
horizontal sectional area in square feet of the vessel at the water 
line is dVidh (after you have done Chap. XVIIL), show that 
A = 2194/i<^'42, and calculate A in each of the above cases. 

Ans. 6840, 6230, 5520, and 4750 sq. ft. 

Ex. 4. Show that in each case A = O'ib x l-42/^» = 0-639/^>. 

Ex. 5. Show that A -^4:20 or lb ^657 gives the diminution or 
increase of tonnage for one inch less or more draught in sea 
water. 

Ex. 6. The water of a dock is such that the divisor of Ex. 1 is 
35-5 ; that is, there is 35'5 cubic feet of it to the ton. What is the 
increased draught of the vessel if her weight is 2098 tons when she 
leaves the sea and enters the dock? Ans. 0*15 ft. 

77. Exercise. For a certain investigation it would be very 
satisfactory if we could express - as a linear function of p ; try if 
this is possible from the numbers given in Art. 75. 

Plotting - and p on squared paper, I find that although there are 
discrepancies we may take 

- = 0-0171 +0-0021i?. 

u 

It depends upon the nature of the investigation what value we 
ought to place upon such a formula. When I use such an approxi- 
mation I always test the importance of the discrepance. 

78. Ex. 1. Plotting the following numbers, we find a regular 
curve. After various trials I found that on plotting x and logy on 
squared paper I obtained a straight line,"^ and so found that 

logy=M955 + 0-0843a;, 

or 2^= 15-69 xlO"«^, 

or y=15-69e-^^. 

* Of course if there is a theory to guide us our labour is greatly saved. For 
example, if we know that the rate of increase or diminution of y with regard 



124 ELEMENTARY PRACTICAL MATHEMATICS 



a; 


5 


6-5 


8 


9-5 


10-5 


120 


13 


14-2 


y 


41-7 


55 


74-3 


loro 


121 161 


196 


247 



If we had many exercises like this there would evidently be a 
saving of labour if we had paper ruled logarithmically one way and 
with equal divisions the other way. 

Ex. 2. Show from the table of Art. 45 [plotting t and logP] 
that, leaving out the census result for 1811, if t is time in years 
since 1811, the population of England and Wales may be said to 
have closely followed the law : 

p _ 1 Q7-03+000556« 

or logP= 7-03 + 0-00556^. 

Calculate P from this formula, find the differences from it as 
given by the census returns, and plot these differences as a curve.* 
What will be the probable population in 1901 and 191111 

Alls. 33-93x106 and 38-58x106. 

Ex. 3. The following numbers were observed in the laboratory : 



T 


0-410 


0-575 


0-895 


1-297 


1-720 


2-200 


2-385 


X 


250 


300 


350 


400 


450 


500 


517 



There were reasons for thinking that T^cx^, where c and k are 



to X is proportional to y itself, it is the compound interest law, and we know 
that y=:ae*^, 

so that we plot x and log y. 

If we know that -J^ is proportional to ^, we ought to try 



If we know that -^ is proportional to .r, we ought to try 
dx 



y = ax'' 
il to .r, 
y = a + bx^. 



* It is to be remembered that we have had three kinds of problems : 

(1) Plotting on squared paper exact numbers calculated from simple formulse. 
Our curve goes exactly through the plotted points, and it is a simple curve. 

(2) Plotting observed numbers : there being errors of observation, we assume 
that the simple curve going most evenly among the points gives us the correct 
law. 

(3) Plotting correct numbers, as of population, the true curve goes exactly 
through the plotted points. But there is possibly a simple law complicated by 
perturbations ; in studying the curve which goes evenly among the points, we 
look for the general law. Having it, we search for the perturbation law, if 
there is one. 

tThis extrapolation published in my 1899 lectures has turned out to be 
wrong. The population of England and Wales in 1901 was 32-5 millions, and 
iu 1911 it was 36-1 millions. 



SQUARED PAPER 



125 



constants. Test whether this is so within the range of the values 
given, and, if so, find the best values of k and c. 

Ans. r= 4-624 X 10- V4'3, 

Ex. 4. Q cubic feet of water were measured as flowing per second 
over a Thomson gauge notch when the difference of levels was // feet. 



H 


1-2 


1-4 


1-6 


1-8 


20 


2-4 


Q 


4-2 


6-1 


8-5 


llo 


14-9 


23-5 



Thomson's theory suggests that the numbers should follow the 
law Q = aH". Try this, and find the best values of a and n. 

Ans. Q=2'^1'2m\ 

Ex. 5. If t seconds is the recwd time of a trotting (in harness) 
race of m miles we have the following published records : 



m 


1 


^ 


' 


4 


5 


10 


20 30 

i 


50 


100 


t 119 


257 


416 


598 


751 


1575 


3505 


6479 


14141 


32153 



Try if there is a law t = am^ Ans. I find t = 1 19m^"'\ 
It is interesting to learn that for all kinds of races of men and 
animals we have t oc m}'^'^^. 



CHAPTER XV. 

IMPORTANT CURVES. 

79. When, instead of a sheet of squared paper, we may use a 
drawing board with T and set squares, the following properties of 
some of the curves mentioned in Chap. XIV. become useful. OX 
and OF (Fig. 18) are the axes of co-ordinates. Let any two angles 



A 


G 


\e 


F 


y^Y 






^^ \ 





D !j L 






P K 


4/ 





Fig. 18. 

XBS and YFQ be set out from any points E and F in the axes. 
Choose any point C. Draw horizontal and vertical lines CA and 
CK. Make BAO = 45°. Make KJD = 45°, project from B and / to 
find the point B* Find other points in the same way. The points 
CEF, etc., all lie upon a curve 

*■ It is easy to show that the angles BA and DJK need not be 45° nor be 
equal to one another. If all the angles FAB, POH are equal to one another, 
each being, say, 60°, and if all the angles RLS, RJK are equal, each being, 
say, 45°, the curve still follows the above law. 



IMPORTANT CURVES 



127 



where the distance OR is h, the distance OP is a, and the angles 
are such that 1 + tan QPY=(\ + tan XRSf. 

Again, in Fig. 19, starting with C, draw CD and CA. The 
Y 




FiQ. 19. 

angles ODG and BAO are each made 45°. The point E is found by 
projecting from G and B. The points C, E, K, etc., all lie upon a 
cur\'e y + a = c{x-{-hy. 

When n is positive, OR is i, OP is a, and 

1 + tan QPY= (1 + tan Zii!^)". 

If a cur^je is given, it is very easy to find whether it follows such 
a law as y = c{x + by\ 

Thus, suppose the curve CEF (Fig. 18) and the axes OX, OV to 
be given ; make OP zero, that is, let PQ start from instead of P. 
Draw CA, AB, and BE as before. From E project to J. Make 
OJK= 45° ; let JK meet CD in ^. Find other points, such as K ; 
see if they meet in a straight line SKR. 

In fact, if either « or i is zero the problem is easy, but if neither 
is zero the test requires some ingenuity. 

Suppose, again, we have made observations of x and y, and we 
wish to test whether they follow the law y = a + ie''^ 

Calculate X= e"" and plot y and X. 

We have to test for y = a + bX% which is the above case. 

It will be noticed that y = a + be""'^" is not more general than 
y = a + be'. 



128 ELEMENTARY PRACTICAL MATHEMATICS 



It would be easy to try for the law y = ^(e^ + c)", as this becomes 
:// = 5(Z+cf whenZ=e^ 

Ex. 1. It is believed that the following observed quantities follow 
the law y = a + bx". Plot them on squared paper; draw the curve 
that lies most evenly among the points. Let CEK (Fig. 19) be the 
curve. In this case the point B is at 0. Draw XES (it will be XOS) 
any angle. From a point C on the curve draw CA horizontally 
and CD vertically. Make OAB = 45° and ODG = 45°. Project verti- 
cally from G to F and E ; make OFM= 45° ; project horizontally 
from E to /and B. Make OJR= 45°. Proceed in this way, finding 
points J5, H^ etc. If the law is true, B^ i?, etc., ought to lie in a 
straight line. Drawing the straight line that lies most evenly 
among them, we find P. The distance OB is - a. 



X 


336-2 


465 


587-4 


628 


813 


y 


651 


871 


961 


1011 


1210 



Ans. ?/ = 210 + 2-21a^-9i4. 

Ex. 2. The following measurements of pressure 'p and volume v 
were made on a gas-engine indicator diagram. It is thought that 
the law is like ^v" constant. 

Test for this by plotting .log^ and log n. If you do not find 



p 


44-7 


53-8 


73-5 


85-8 


113-2 


135-8 


V 


7 03 


5-85 


4-30 


3-50 


2-50 


1-90 



a straight line it may be that there is a constant error in ?; due to 
wrong measurement of the clearance. Try therefore by the above 
method for the law « = c (v + a) ~ ". 

Plot V horizontally as x in Fig. 18, and v vertically. Take the 
point P dX and proceed as directed. 

Ans. I find points like SK^ etc., lying in a straight line. Drawing 
this line I find B and (?i^ = 2-10. 

Ex. 3. In the last exercise, increase every v by the amount 2-10 
and call it u. Plot log^:> and log % on squared paper and try for 
the law. I find i?('y-}-2'l)^'^^^ = 900. 

Ex. 4. It is suspected that the following quantities observed in 
the laboratory follow the law 



X 


05 


Kj 


1-0 


1-2 


1-4 


1-6 


1-8 


20 


y 


3-730 3-912 


4-325 


4-709 


5-242 


5-953 6-927 


8-213 



IMPORTANT CURVES 



129 



First compute the values of e" and call them X. Now plot 
y and A" and draw the best curve. Proceed as in Fig. 19, making 
OB = 0. I find OF to be -3-2. I would now write out a new 
table of y and X as corrected by the curve, and I would plot 
\og{ij + S-2) and logX These ought to give a straight line. I 
find the answer y=.3'2 + 0'25e^-^\ 

80. If three points only are given and I am asked to join 
them by means of a curve, and I do not know anything about 
the nature of the curve except that it ought to be simple, I 
usually make a choice between 

y=^a + bx + cx^, (1) 

y = a + bx", (2) 

y = a + be'"' (3) 

Exercise. Given the three points 



X 


1-5 


3-5 


5 


y 


6-24 


16-45 


27-07 



Let (1), (2) or (3) be the equation to the curve passing through 
the three points and find the values of the constants. 

An8. y=l-54 + 2-284.'C + 0-5643.7;2, (1) 

?/ = 2-742-i-l-823i»^^ (2) 

'y= - 15-83 + 16-57e<'^'^" (3) 

To make (2) or (3) pass through three points needs a squared 
paper method of solving an equation. Take for example the 
more tedious form (3) Let e* be called X. Let Xj, Xg, and Xg 
correspond to the three given values of x, and let y^, y^^ and y^ 
be the three given values of y. 

y^ = a-{-hX^% 

y^ = a^rhX^% 

y^ = a + hX^\ 

Subtracting to get rid of a and dividing to get rid of h, we have 

fe-y, 10-62 X.'-AV 

Clearing of fractions and dividing by Xj", we have 
g:j-5o _ 2 -04026^ + 1 -0402 = 0. 

By trial and the use of squared paper, I find c = 0190, and it is 
easy to find 6 = 16-57, a = - 15-83. 
P.M. I 



130 ELEMENTARY PRACTICAL MATHEMATICS 

81. Use and Misuse of Empirical Formulse. As a result of 
experiments, the following corresponding values of x and y were 
obtained : 



X 


4 i 5 


6 


7 1 8 


9 


10 


11 


y 


6-29 


5-72 


5-22 


4-78 


4-39 


4-06 


3-75 


3-48 


y' 


6-28 


5-66 


515 


4-72 


4-35 


4 05 3-78 


3-55 


y" 


6-23 


5-72 


5-26 


4-83 


4-44 


4-08 


3-75 


3-45 



These being plotted on squared paper and tested in various ways, 
it was found difficult to say which of the two empirical formulse, 

57-12 .,. 

y=^YVx (^) 

or y = 8-7l€-«°««^, (2) 

more exactly represented the observed numbers. The values given 
by (1) are tabulated as y\ and those given by (2) are tabulated 
as y'\ and it will be seen that the discrepancies are much the same 
for both the formulae, and in no case are they large. 

It is not unusual for the man who discovers an empirical fomiula 
to use it for extrapolation. Now, if we use (1) or (2) for such a 
purpose, calculating y for aj = 1 or a? = 20, note the difference in 
the answers : 



X 


1 


20 


y' 


9-36 


2-28 


y" 


8-005 


1-61 



Either formula is, therefore, good enough for calculating y inside 
the experimental range ; and good enough, no doubt, for many 
purposes, but it is evident that extrapolation is dangerous. 

[The following remarks will be better understood after the 
student has reached Chapter XVIII. ] 

Furthermore, (I) and (2) represent quite different laws : 



^ from (1) is proportional to y^ whereas 



dx 



from (2) is 



proportional to y. 

When, therefore, a man smooths his plotted points, drawing the 
curve which lies most evenly among them, it is most important to 



IMPORTANT CURVES 131 

know the law of smoothing, for if he uses different kinds of curves 
he may obtain very different results. All of them may be accurate 
enough for some purposes and wholly misleading for others. 

When there is no guiding from theory, an empirical formula 
may be adopted, deduction from which may be quite misleading. 
Again, when we have a hypothesis of which we desire to make a 
theory to fit the facts, it is often much too easy to do so. 

It is of course mainly in finding -^ or the higher differential 

coefficients that empirical formulae are dangerous. 

When men smooth their observations by means of a curve they 
are really doing the same thing as if they used an empirical formula. 

I have not tried, but I should think that the three formulae of the 
exercise in Art. 79, which probably all give nearly the same values 
of y for any value of x between 1 '5 and 5, give quite different values 
of y for such values of a; as 1 or 10. 



CHAPTER XVI. 

SQUARED PAPER. 

82. When, as the result of an investigation, we have arrived at a 
formula connecting y and x, we sometimes try to replace it by an 
approximate simpler formula. 

Ex.1. Plot 2/ = 9-9 + 0-6167a; + 0-35a;2-^«;3 (1) 

Find the nearest approximate linear law between the values 
a; = 0*5 and .'r = 3-0. 

Ans. ,y = 9-7 + M4a; .' ..(2) 

Of course the student sees that he takes values of x, in each case 
calculates y, and plots (1) on squared paper. He then finds the 
straight line which lies most evenly among his points, and so 
finds (2). This simple formula must not be used for values of x 
outside the range 0-5 to 3*0. 

Ex. 2. In my theory of Struts it simplified the reasoning 
enormously — in fact, without this, reasoning was practically im- 
possible — to be able to use 

a , 1 • 

= — r- and 7= 

i~-ox cos V« 

as if they were the same. 

Let — be called y. For values of x from to 0*9 calculate y 

cosvx 
and tabulate. We must try if y is approximately represented by 
the other formula or ^ 



'-l-hx' 
that is, y - hxy = a. If, now, we plot y and xy as the co-ordinates of 
points on squared paper, or if we plot x and -, we ought to get 

(approximately) a straight line. We choose the best straight line 
and calculate a and fe. Ans. a = 1*003, & = 0-471. 



SQUARED PAPER 133 

Ex. 3. Ill Art. 26 we found that if n is the number of years in 
which a sum of money will double itself at compound interest at 
r per cent, per annum, then 



= l0g2H-l0g(l+j^). 



Take various values of ?•, say 2, 2^ 3, 3^, 4, 4i, 5, and in each 
case calculate n. 

Now, to find if there is not a simple approximate law connecting 
n and r, try various systems of plotting, such as we have tried in 

other cases. If you plot ii and -, you will find your points lying 

so nearly in a straight line that you know you may use the approxi- 
mate formula ^^q 

r 
if r is not greater than 5. (See also Ex. 5, Art. 113.) 



Ex. 4. Plot ^?/=l+Vl+«2. 

Within what limits of values of x may we take 
y= 1-23 + 0-983; 
as equivalent to it. Ans. From iK = 2 to «= 10 or more. 

It may be deduced from this, that we may take a + \/a^ + x^ as 
nearly equivalent to 

l'23a-f- 0'98a:, 

between the limits x = 2a and x = lOrt. 

83. Any equation in x, however complicated, may be solved. 

Ex. 1. Find a value of x which will satisfy the equation 
4-3a^-^ - 5-4e-°=^ + l7a;-=^"^ - 12 = 0. 

To solve this equation, let us say that 

,j = 4-3.>'=^-^ - 5-4e-*''=^ + \1x-^' - 12. 

Take various values of ,/', calculate the corresponding values of //, 
and plot on squared paper. It is easy to obtain from this an 
approximation to the value of x which will give y = 0. This leads 
me to take x=\'2^ and I find ;y = 0-280. Now I try a; =1*25, and 
I find 11= -01 40. Hence the correct value of x is between these. 
Plotting these points to a large scale and joining by a straight line, 
I see that a: = 1-23 is probably right, and on trial I find it sufficiently 
correct for all practical purposes. By trying 1231 and 1*229, and 
plotting the results to a still larger scale, I can obtain even a closer 
approximation ; and, indeed, there is no limit to the accuracy with 
which one may solve such an equation but that due to want of 
accuracy of the tables used in calculation. 

If there are several values of x which will cause y to be 0, of 



134 ELEMENTARY PRACTICAL MATHEMATICS 



course they also are answers. Every value of x which satisfies an 
equation is called a root of the equation. 

Ex. 2. The following equation may be very easily solved in 
another way, but the student ought to solve it as an illustration 
of our squared-paper method of working. 

.'^2 - 5-1 IxH- 5-709 = 0. 

Let2/ = a;2-5-ll« + 5-709. 

Taking the following values of x, I have calculated the values of y : 



X 


' 


1 


1-5 


2 


25 


3 


3-5 4 


5 


y 5-709 


1-599 


0-294 


-0511 


-0^816 


-0-621 


0^074 


1-269 


5-159 



Plotting these on squared paper, the axes being OX and OY, I 
get a curve. 

Notice that it cuts OX in two points, P and Q, so that there are 
two roots to the equation, OP and OQ. 

By closer approximation, as in the last pase, near x = 1^64 or r66, 
and «=3^45 or 3-47, I find that the two roots are 1-65 and 3 •46. 
Solving the equation in the usual way (see Chap. V.), I find these 
same answers. 

How many advanced students can solve even a cubic equation ? 
In practical work we may have to solve any equation whatsoever, 
and we now have a method of doing this with any amount of 
accuracy that may be desired. 

Ex. 3. Find a value of x to satisfy 

5.3^01040: g|^2 Q.g^ _^ 0-78a;^-^2 ^.^g a; _ 2-126 = 0. 

The student must remember that 0'd)X is in radians, and must be 
multiplied by 57^296 to convert it to degrees. Ans. «- = 0"74. 

Ex. 4. Find the value of x for which 
tan a? = 2 •46a'. 

Let y = tan « - 2 "460;. 

Taking a few values of x at random, and appealing occasionally to 
squared paper, one is able to get an approximation quickly. Of 
course, an angle x radians is 57-296 x degrees. 



The horizontal lines show the places during the 
calculation at which I appealed to squared paper 
to assist me in my next guess. Ains. x= 1-258. 



X 


y 


1 

1-5 
1-3 


-•903 
+ large 
•4038 


1-2 


- -381 


1-25 
1-26 


-•066 
+ •015 



SQUARED PAPER 135 

Ex. 5. Find in each case a value of x which satisfies each of 
the following equations : 

2a;2-5_5a;-8 = 0, Am. «=2'55. 

«2 - 10 logio a; - 3 = 0, Ans. x = 2'1. 

a;3-2«2- 400 = 0. Am. 3; = 8-10. 

Ex. 6. Find the three values of x which satisfy 
ic3 - 157.2 + 61x- 75 = 0. 

Ans. 3, 2-822, 9-178. 
Ex. 7. Find a value of x which satisfies 

g\/^+logioX' = 0-8558. Ans. a; =31 62. 

Ex. 8. Find a value of x (in radians) which satisfies 

sin2rc + 5sina;+3a;=l-38. Ans. a: = 0-1386. 

Ex. 9. In Chapter XV., to work another problem we had to 
find a value of c which satisfies the equation 

g3oc_ 2-0402 e2c+ 1-0402 = 0. Ans. c = 0-190. 



CHAPTER XVII. 



MAXIMA AND MINIMA. 



84. Ex. 1. Divide the number 10 into two parts such that the 
product is a maximum. 

Let X be one part, then 10 - a; is the other. 
Let the product be called y, or 

y = x{lO -x) or \Ox-x^. 

Take various values of x, calculate y in each case, and plot on 
squared paper. The maximum value of y is evidently given hy x = 5. 

Ex. 2. When is the sum of a number and its reciprocal a 
minimum 1 Let x be the number ; when is 

1 

y = x + - 
^ X 

a minimum? Taking the following values of x, we find y in each case : 



- i 


3 

4 


1 


4 


2 


y 


2i 


•2tV • 


2 


2' 


'A 



On plotting, we see that ic = 1 is the answer. 

Ex. 3. What is the greatest cylindric parcel to be sent by parcel 
post"? The Post Office regulation is that length plus girth must not 
be greater than 6 feet. Let x be the radius of the circular section, 
then 6 - lirx is the length. The volume is 

V^ttX^I^^-Ittx). 

What value of x will make v a maximum'? Taking ic = 0-4, 0*5, 
6, 0*7, 0*8, etc., and in each case calculating «;, plotting on squared 
paper, we find that a; = 0-636 feet. 

Ex. 4. Divide 10 into two parts such that three times the 
square of one part plus four times the square of the other shall be 
a minimum. 



MAXIMA AND MINIMA 137 

If X is one part, 10 - ic is the other. When is 

a minimum'? Calculating and plotting, we find aj = 5-71, and the 
other part 4-29. 

Ex. 5. A man at A is at sea 4 miles distant from the nearest 
point of a straight shore. He wishes to get to a place B^ which is 
10 miles distant from this nearest point. He can row and walk. 
Find at what point D he ought to land to get to B in the minimum 
time, if he rows at 3 miles per hour and walks at 4 miles per hour. 
Assume that he can equally well leave his boat at one place as at 
another. Here ^(7=4, (7^=10; let CI) = x, then AB^slW^x^ 
and DB= 10 - a;, so that the whole time in hours is 

y = ^v/l6T^2 + l(iO-a:). 

Calculate y for various values of x; plot and find that x = 4*535 miles 
gives the minimum time, 

Ex. 6. The strength of a rectangular beam of given length, 
loaded and supported in any particular way, is proportional to the 
breadth of the section multiplied by the square of the depth. If a 
cylindric tree is 15 inches in diameter, what is the strongest beam 
that may be cut from it % Let x be its breadth. Draw a rectangle 
inside the circle and you will see that the depth is \/225 - a;^. Hence 
the strength is a maximum if ?/ is a maximum, where 

y = a;(225 - a;2) or ^ = 225^-0:^.... 

Calculating y for various values of x and plotting, we find that 
the strength is a maximum when the breadth x is 8 66 and the 
depth is 12 25 inches. 

Ex. 7. The stiff"ness of a rectangular beam of given length, 
loaded and supported in any particular way, is proportional to the 
breadth and the cube of the depth of its section. If a cylindric tree 
is 1 5 inches in diameter, what is the stiffest beam that may be cut 
from if? If X is the depth, the breadth is v/225 - x?-, and we desire 

^^^^ y = x?J^M~^^ 

shall be a maximum. Calculating y for various values of x and 
plotting, we find that the stiffness is a maximum when the depth x 
is 13 inches and the breadth 7 5 fnches. Observe the difference 
between this and the answer to Ex. 6. 

Ex. 8. When a certain vessel moves through the water at x knots 
the total cost in wages, depreciation, and interest on capital, stores, 
coals, etc., in pounds per hour, is 4 + 0'001ir3. Yov a passage of 



138 ELEMENTARY PRACTICAL MATHEMATICS 

1000 miles, at what speed ought the vessel to go if the total cost on 
the passage is to be a minimum? Here the number of hours is 

, so that the total cost is 

X 

i5^(4 + 0-001a:3) or 100o(- + 0-001ic2\ 

Let y = - + 0-001a;2. 

^ X 

For various values of x calculate y and plot. The answer will be 
found to be 12*60 knots. 

Ex. 9. If V, the speed of water in a river, is 4 miles per hour, 
and x is the speed of a steamer against stream relatively to the 
water ; if the total cost of the steamer per hour, including coals, is 
0*5 + O'OOOSa:^, find the speed x so as to make the cost of an upstream 
passage of 100 miles a minimum. 

The speed of the steamer relatively to the bank is ic - -y ; the time of 

the passage is hours, and hence the cost of the passage is 

100(0-5 + 0-0003a;3 ) ^^ 100(0-5 + 0-0003a;3) 

x-v X-4: 

J ,.. 0-5 + 0-0003a:3 , , .. , . , . 

Lettmg ?/ = — — — — , calculatmg y for various values of x 

X — ^ 

and plotting, it will be found that the best x is about 8*5. That is, 

the speed of the steamer relatively to the bank is 4*5 miles per hour. 

Exercises of this kind can be worked generally algebraically (see 

Chap. XXII. ), but this plotting method of working ought always 

to be resorted to also. It tells us what is the extra loss if the best 

speed is not adhered to, and very often the extra loss is not great. 

Ex. 10. In a submarine cable, if d is the diameter of the copper 
wire and D is the diameter of the gutta-percha covering; the distance 
to which readable signals may be sent is greater as 

y=dnog^ 

is greater. Take i> as 10; for various values of d calculate y and 
plot on squared paper ; what value of d is best 1 It is quite evident 
that we can use common logarithms. Atis. d = 6'0Qb. 



CHAPTER XVIII. 



THE INFINITESIMAL CALCULUS. 



Thus ^^ is the sine 



85. Slope of a Straight Line. In Art. 60 we saw that y = a + bx 
is a straight line. 

I have called b the slope of the line. We saw that if x increases 
by the amount 1, y increases by the amount b. 

Let CD be the line ; the distance OC represents a. P and R are 
two points in the line, and if PQ is 1, then QPi is b. 

b is the rise for 1 horizontal. 

Note that when we say that a road rises -}^ or 1 in 20, we mean 
1 foot rise for 20 feet along the sloping road, 
of the angle of inclination of the y 
road to the horizontal. Our slope is 
different, being the tangent of the 
angle PiPQ. [For the meanings of 
such terms as sine and tangent the 
student is referred to Art. 34.] Look- 
ing upon ?/ as a quantity whose value 
depends on that of x, observe that the 
rate of increase of y relatively to the in- 
crease of X is constant, being indeed b, 
the slope of the line. The symbol used 

for this rate is / 
ax 

Try to recollect the statement that if y = a + bx, then ^-. = ^, and 

that if -r- = b, then it follows that y = A+bx, where A is some 
ax 

constant or other. 




Note that it is one symbol; it does not mean 

dy 



dxy 



Exercise. If 4:X + 3y=7, find ^« 



140 ELEMENTARY PRACTICAL MATHEMATICS 



Here 



y = 3-3.- so that ^ 



Now, if a road is of perfectly constant slope, it is easy to measure 
this slope. It rises 1 foot, say, for 10 horizontal, or 2 feet for 

20 horizontal, or 3 feet for 30 horizontal. ^ is 0*1. But if the 

dx 

slope of a road is continually altering, and we want to know 
the slope at a particular place, we really only measure an average 
slope, however small our distances may be. 

86. In the curve of Fig. 21 there is a positive slope (y increases 
as X increases) in the parts AB^ CD, EF, GH, and negative slope 

Y 




Fig. 21. 

{y diminishes as x increases) in the parts DE and FO. The slope 
is zero at D and F, where y is said to reach maximum values, and 
it is also zero at E and G, where y is said to reach minimum values ; 
it is also zero at i?, which is neither a point of maximum nor 
minimum. It must be evident to a student that the slope of a 
curve at a point is the slope of the tangent to the curve at that 
point. Suppose we want to know the slope at the point P (Fig. 21). 
Imagine Fig. 22 to be a great magnification of the curve at point F. 
If PD or 01 is x and PI is y, let F be another point and let FC 
or OH he x + 8x and FH be y + 8y. The student must note that Sx is 
a symbol for a small increment to «; a new kind of symbol. It 
does not mean a quantity 8 multiplied by a quantity x. Then PG 

is Bx and FG is Sy, so that -y-^ or / is the average slope from P to F, 

FG ox 

and this is really tan FPG. 

Now let F be nearer P, say at F', or nearer still, at F". In every 

case the average slope is the tangent of the angle made by FP 



THE INFINITESIMAL CALCULUS 



141 



Or F'P or F"P with the horizontal. But it is not until 83/ and Sz 

are imagined to get smaller and smaller without limit that ^ can be 

ox 

called the actual slope at P. The line joining F and P or F' and 




I H" H' H X 



Fig. 22. 



P or F" and P gets to be more and more nearly what we call the 
tangent at P as i^ or F' or F" gets nearer and nearer to P. 

When we speak of hx getting smaller and smaller and -^ reaching 

a limiting value, we distinguish this limiting value by the 

symbol -^, and it is evidently tanPJ5X if PB is the tangent to 

the curve at P. I do not think that anybody can draw a tangent 
to any curve at a given point very accurately, but if he could he 

would have no difficulty in finding -^ at the point. It is a pity 
that the word tangent is used for two different things. ^ is the 

tangent of the angle that the tangent to the curve makes with 
the horizontal. 

In Fig. 21 the slope at P is tan PiOT, and is positive. The slope 
at Q is tan QJX^ and the tangent of an obtuse angle is negative. 

If, in Fig. 21, PL = y is, say, population, and OL = x means 

time after some date ; the slope at P is .^ , the rate at which the 

population is increasing per annum. The scale to which this rate is 
represented by tan PKL or the slope, is easily found, because the 
rate is the populatio n represented by PL 

* time represented by KL 



142 ELEMENTARY PRACTICAL MATHEMATICS 

87. If s is the distance in miles that a train has reached from 
Euston at the time t, how do we study the motion ? If a Bradshaw's 
railway guide gave, not merely the time of reaching a few stations, 
but the time of reaching say fifty places between London and 
Northampton, the record would be something like this, only the 
student may imagine many more entries ; entries perhaps for every 
half-mile 









t 


s 


Hours and minutes. 




Time 


in hours since 


Distance in miles 






leaving Euston. 


from Euston. 


10.20 o'clock 












10.32 „ ... 






•2 


6 


10.35 „ ... 






•25 


10 


10.40 „ ... 






•33 


14 


10.44 „ ... 






•40 


14 


10.47 „ - - - 






•45 


18 


10.53 „ ... 






•55 


24 


etc. 






etc. 


etc. 















































:5 






















































































































^ 




































/ 






































^ 


/ 














20 




















/ 


/ 


^ 

i^ 


































/ 


/ 






















10 












/ 


^ 


H-> 


/ 
































/ 






































y 


/ 



































•2 



•4 -5 -6 

Fig. 23. 



•8 Hours 



If we plot 5 and t on squared paper as in Fig. 23, it is* evident 
that the slope of this curve everywhere represents the speed of the 



THE INFINITESIMAL CALCULUS 



143 



train. See how the speed quickens on leaving Euston, until from 
s = 5 to s=ll miles it keeps nearly constant, but diminishes and 
becomes at 5= 14 miles. In fact, the train has stopped ; then we 
see the speed getting up again, then diminishing and increasing 
until at s = 24 there is a crawl for a mile. 

If we represent the train as coming back again, of course, there 
will be negative slope, as the velocity is negative. 

Ex. 1. If s and t are carefully plotted, the slope of the tangent, 
which can be carefully drawn at every point, shows the speed ; but 
tangents are not easy to draw with accuracy. The following 
method is better : — First plot carefully. Now draw a curve through 
the plotted points. Tabulate the values of s for equidistant values of t. 
Here is a particular case. In a new mechanism it was necessary for 
a certain purpose to know in every position of a point what its 
acceleration was. A skeleton drawing was made and the positions 
of the point marked at intervals of time from a time taken as 0. 
In the table I give at each instant the distance of the point from a 
fixed point of measurement, and I call it s feet. 



t 

Seconds. 


Feet. 


V 

Feet per second 
or hsjU. 


Acceleration in 

feet per second 

per second or fi»/S«. 


•06 
•07 
•08 
•09 
•10 
•11 
•12 
•13 


•0880 
•2354 
•3703 
•4925 
•6020 
•6986 
•7821 
•8525 


14 74 

1349 

12 22 

10-95 

9 66 

8^35 

7-04 


-125 
-127 
-127 
-129 
-131 
-131 



If a body of 200 lbs. has this motion, what force at every instant 
must be acting upon it to maintain the motion 1 

The mass in Engineers' units is 200^32*2, or 6*2. 

Multiply 6^2 by an acceleration 127, and we find the force to 
be 787 lbs. 

When in the above table I find a velocity of, for example, 
14^74 feet per second, remember that it is the average speed 
from / = 0^06 to ^ = 0*07; I have assumed that it is the real speed 
at t = 0-065. But this, generally, can only be approximately correct. 
I have found the assumption to be accurate enough for many 
practical purposes. There are obvious tedious ways of getting 
better approximations. 

But I must consider what is the velocity at a particular instant 
more carefully, and I therefore take up the following exercise. 



144 ELEMENTARY PRACTICAL MATHEMATICS 



88. Exercise. The motion of a train leaving Euston is such that 

s being in miles from Euston, t in hours from leaving Euston. 

At the end of 01 hour, find v, the velocity of the train in miles 
per hour. 

Take ^ = 0'01, 0*02, etc., and in each case calculate s. 

Plot on squared paper. I get the curve OQPR (Fig. 24.) At P, 

^ = 0*1 hour, .s = 3 miles; drawing 
the tangent PK at P, I find that 
in the scale of time KL represents 
0*05 hour and in the scale of dis- 
tance PL is 3 miles. Hence the 

slope at P represents qTqI^^, or 

60 miles per hour. As a matter 
of fact, this is the correct answer, 
but I did not get it by drawing 
the tangent. Nobody could draw a 
tangent so as to get the perfectly 
correct answer. I got the answer 
in the following way. I calculated 
s' for the following times : 

8^ is the excess timebeyondO'l hour. 

8s is the distance in excess of 3 

miles. 

Ss . 

J- is the average speed during the interval U, the space Bs being 

passed over in this interval. 




K L 

Fig. 24. 



Hours 



t 


s 


St 


8s 


Ss 


•1 


3 


8t 


•11 

•101 
•1001 


3-63 
3 0603 
3 006003 


•01 

•001 

•0001 


•63 

•0603 

•006003 


63 

603 

60-03 



It is evident that as we take 8t less and less we are getting nearer 
and nearer to the actual speed at the time 0*1. 

To be quite sure of this let the work be done algebraically. 

At the time t calculate s ; again, for the new time t + 8t calculate 
s-\-8s. 

We have 

.9 = 300^2^ and s + 8s = 300 {t + Sty, or s + Ss = 300{t^ + 2t . 8t + (8ty}. 
Subtracting, we get 

8s = 600t. 8t + 300 {8ty. 



THE INFINITESIMAL CALCULUS 145 

Now divide by St, and we get average speed 

^ = 600^ + 300.8^. 
or 

Please notice that this is exactly true for any value of St. Now 
I come to the important idea; as St gets smaller and smaller, 

— approaches more and more nearly 600/, the other term 3008t 
St 

becoming smaller and smaller without limit. 

Creatures of our limited senses and faculties cannot possibly 
comprehend infinity, but, for mathematical purposes, when we say 
that a thing is infinitely great, we mean that the thing is greater 
and greater without limit. 

When we say that a thing becomes smaller and smaller without 
limit, we mean that it becomes ; that is what we mean by nx)thing. 

Hence in the limit — is truly 600/. The limiting value of - as 

St 7 Of 

5/ gets smaller and smaller is called -=- or the rate of change of 

(ti 

s as t increases, or the differential coefficient of s with regard to /, 

or it is called the velocity at the time t. It is only when St is made 

smaller and smaller without limit that we speak of the actual 

ds 
velocity -j or make the statement : 

ds 
Actual velocity at time / is -^i = 600/. 

This formula enables us to find the velocity at any time. When 
/ is 0-1, the velocity is 600 x 0-1 or 60 miles per hour. 

Now, surely there is no such great difficulty in catching the idea 
of a limiting value. Some people have the notion that we are 
stating something that is only approximately true; it is often 
because their teacher will say such things as " reject 3008/ because 
it is small " or " let dt be an infinitely small amount of time," and 
they proceed to divide something by it. Men like this are incapable 
of acquiring common sense. 

Another trouble is introduced by people saying, "Let St = and 

- or — is so and so." The true statement is, " As St gets smaller 

St dt g, 

and smaller without limit, ^^ approaches more and more nearly 

0/ 

the finite value 600/," and, as I have already said, everybody uses 
the important idea of a limit every day of his life. 

The student ought to give a good deal of thought to this idea 

of a limiting value of ^ as St is made smaller and smaller. Of 

Of 

course Ss gets smaller and smaller also. 

He must not say, " Let Ss be and let St be 0," because divided 
by may mean anything. Let him consider carefully what he 
P.M. K 



146 ELEMENTARY PRACTICAL MATHEMATICS 

means by velocity. If a body moves uniformly through 1 foot 

in 1 second it moves through O'OOl foot in 0-001 second, and it 

moves through the millionth of a foot in the millionth of a second. 

The ratio of the space to the time is the velocity, and if velocity 

is changing, the velocity at any instant can only be found when 

we measure an excessively short space and the excessively short 

time in which it is described. 

The plain man of common sense finds no difficulty in catching 

the idea. Two thousand years ago neither he nor a small boy 

would have had a difficulty in understanding that a hare would 

beat a tortoise in a race; it is the mathematical philosopher who 

makes a difficulty about such matters, and in these days he says 

that this fundamental idea of the calculus can only be comprehended 

by a mathematician. This would not matter if these philosophers 

were not entrusted with the education of youth, a trust for which 

all their training has unfitted them. When they come to explain 

8s 
the essential idea of the limiting value of — they talk foolishly. 

Of 

89. Let me give the idea in another form. Suppose y and x to 
be any quantities whatsoever, connected by the law 

y = ax^ (1) 

Take a particular value of x and calculate y. Now take a new value 
of X, say X + 8a;, and calculate the new y, calling it y-\-8y ; 
y + 8y = a{x + 8xY, 

or y + 8y = a{x^ + 2x.8x + {8xy} (2) 

Subtracting (1) from (2), we have 

8y=2ax.8x + a.{8xY; 
divide by 8x, and we have 

^£=2ax + a.8x (3) 

Now (3) is true for any value of 8x. It gives the average rate of 
increase of y in regard to x. But it is only when we let 8x get 
smaller and smaller without limit that we say 

1=2-' w 

and the actual rate of increase of y with regard to x is known for 
any value of x. 

90. Similarly, if y = ax^, (1) 

y + 8y = a{x + 8x)^, 
y + 8y = a{x^ + 3x'^ . 8x + 3x . {8x)^ + (8xf} (2) 



= na9f~\ 



THE INFINITESIMAL CALCULUS 147 

Subtracting (1) from (2), and dividing by 8x, we get 

^ = 3ax^ + Sax . 8x + a{^xf. 

Now, letting & get smaller and smaller without limit, we use the 

new symbol . 

J = Sax\ 
ax 

91. With a little more knowledge of algebra than I presume in 
my hearers, it will be found easy to prove (see Art. 95) that if 

y^ax"", 

where a and n are any numbers whatever, positive or negative, 

dj 
dx 

This rule comprehends the others £hat I. have given. In fact, if 

y = a + bx + cx^ + etc. + mx'\ (1) 

-^=^0 + b + 2cx + etc.+nmx''-'^ (2) 

If y is given as a function of x, we are said to differentiate when 

we find -/. 

ax 

92. In some of the above exercises we had y given as a function 

of X, and we were asked to find -^. We see that this rule is very 

dx 

much easier to apply than any rule that requires us to plot the 

curve on squared paper. -^ is sometimes called the slope of the 

ax 

curve, or the tangent of the angle which the tangent to the curve 

makes with the axis of x, or it is sometimes called "the Differential 

Coefficient of y with regard to x," or "the rate of increase of y with 

regard to x." 

93. Sometimes we are asked to Integrate ; that is, when given 

^ we are asked to find y. 
dx 

Thus given (2), above, we may be asked to find (1). 

The integi-al of a is ax, the integral of /3x is ^^x^, the integral 

of yx- is lyx^, and, generally, the integral of aa;"* is -a;"*^^ for any 



148 ELEMENTARY PRACTICAL MATHEMATICS 

value of m."^ When we integrate we add a constant, which may be 
any constant, because the differential coefficient of a constant is 0. 

Ex. 1. Differentiate 

y = i-\-Zx + 0-7^2 + 2-15;7;3 + 15a;20 + 12«-i + 24;r-3-o^ + 23:0-786. 

Ans. ^i^ = + 3 + 1 -4.^ + 6 •45.7;2 4- 300.^19 - 1 2.^ - 2 
ax 

-72-96a:-404+i.572a:-o-2H 

Ex. 2. Integrate 3 -056 + 2.'^ + 1 5.7:2 + 1 2x^^ + 1 -bx " 2 

+ l7a;-o-46 4-2a:4-567. 

Ans. C+3-056.'c + a;2 + 5^3 + 3_^i6_ 1.533-i4.l7.8,^o-954 + 0'3592a:5-567. 

The symbol ,^ means the differential coefficient of ^^ . The 
differential coefficient of 1-^ is indicated by y^- 

* This rule fails when m= - 1. The integral of ax~'^ is a logeO?. See Art. 95. 



CHAPTER XIX. 
FORMULJE AND PROOFS. 

94. When the student knows how to diiferentiate and integrate 
«" he can utilize the calculus in most practical engineering problems. 
The ordinary student spends months in differentiating and in- 
tegrating all sorts of curious expressions for which he seldom has 
any practical use afterwards. Besides a;", it may here be worth 
while to give a few others. 





dy 




The iutegi-al of u, 


y 


dx 


u 


written u . dx. 


a constant 








a any constant 


ax 


a 


b 


bx 


ax' 


"lax 


bx 


ibx^ 


ax'' 


nax'^~^ 


bx"' 


ni + 1 


log a; 


1 

x 


1 

X 


log a; 


log (u; + tt) 


1 
x + a 


1 

x + a 


log (x + a) 
a 


fee** 


abe'*'' 


6e«' 


A sin (ax + 6) 


aA cos [ax + h) 


jBcos(aa;-i-&) 


sin (ax + b) 


A cos (aa; + 6) 


-aA &va.{ax + h) 


Bsiniax + b) 


cos {ax + b) 



In future, log x will always mean the Napierian logarithm of x ; 
the common logarithm will be written logj^a;. 

Any other letters than x and y and u may be employed. 

95. The only proofs which I can give for the present are these : 



150 ELEMENTARY PRACTICAL MATHEMATICS 



or 



y + ^y=af'-{-n.^x. .y>'-i + ^ ^ ^ {^xfx"-'^ + etc., 



by the Binomial Theorem. See Art. 28, Ex. 5. 

Subtracting, ^y = n. 8x . x"--^ + ^^y^^ (Sxf . x"-2 + etc.. 

Therefore ^ = 7a;"-i + ^^Lzi) s^. . ^n-2 _^ etc. 

0^ 1.2 

Now let 6.1' get smaller and smaller without limit ; we see that all the 
terms which have 8x or (8x)^ or higher powers of 8x become zero, and so 

ax 

2. If y—e^. I have tried in many ways and failed to find a proof of 
our rule that does not involve the exponential theorem. 

Here is the simplest form of this theorem (see Art. 28, Ex. 6) : 

3^ = ^ = l+^ + _ + _ + _ + etc. 
Differentiating this term by term, we find 

|=0 + l + x+|+g + etc. = e'=y. 
In the same way we can show that if y=e"*, then 

•ax 

3. If y = log.^•, then x = ^, — = 6^=^, 



or 



dy 
dx X 



4. If y = sin.r, y + 8y = sin(^+8^), 

8y = sin(^' + 8x^ — sin x. 
In elementary trigonometry it is shown (see Ex. 3, Art. 28) that this is 
8y='^ cos (^ + ^ 8x) sin ^ 8x. 

Hence |=cos(:.+i8^)^i|M-. 

It is easy to see by drawing a small angle a and recollecting what 
sin a and a are, that sin a-i-a becomes more and more nearly 1 as a gets 
smaller and smaller. Hence sin ^8.^•-^^8A• becomes 1 in the limit, and 

dy 

-~- = cos X. 

dx 

We can in the same way show that if y = cos x^ then 

dy 

dx 

See Chapter XXXII. for an extension of these rules. 
When preparing this course of lectures for the press I discovered new 
easy proofs of (1) and (2) ; they are given in Chapter XXXII. 



CHAPTER XX. 

THE CALCULUS. 

96. I will now give an example of the usefulness of integration. 
Y 




Let BPCD be any curve; at the point P let FP or OE be called x, 
and PE be called y. 

Let the ordinate QE of the curve GQJ represent A, the area of 
HPEG. I have taken HG anywhere as my starting ordinate from 
which the area is to be counted. Let OE' be called rc + &-, so 
that EE' = 8a;. 

Now QE = A represents area HPEG, 

Q'E' = A + 8A represents area HP'E'G. 

Hence 8 A is the area PFE'E. 

But as hx is smaller and smaller, we find it to be more and more 
true that g^ _ pj^ ^ g^ _ y g^. 

dA 



Hence 



dx 



= «/• 



(1) 



152 ELEMENTARY PRACTICAL MATHEMATICS 

That is, the ordinate of a curve is the differential coefficient of its 
area, or area A oia curve is the integral of y, the ordinate of the curve. 

97. Exercise. Let 0PM (Fig, 26) be the curve y = ax^, a curve 
which is known as a parabola. 

Then, integrating, A^^x^ + C, 

where C is some unknown constant which depends upon where we 
count area from. 



Q 


Y 


P 


I 


H 






( 


3 G 


I 


I X 



Thus if we count area from x = 0, that is, let ^ = when x = 0, 

then C=0, so that A=-^x^. Here we have a formula which gives 

us the area of the curve up to any value of x. 

Again, find the area between the ordinates HG and FE. 

A up to FE from some ordinate = ^ OE'"^ + C, 
A up to HG from the same ordinate = ^ OG^ + C. 

Hence area required = ^ {OE^ - OG^). 

Suppose we want the above areas in terms of FE, etc., instead 
of a. As y = ax\ and hence FE = ax OE^^ or a = FE/OE^, we can 
replace a in the above formulae. Thus 

area of FOE = %OE^=\ ^,xOE^--^\fE . OE 

= K of the area of the rectangle QFEO. 

The proof of Simpson's rule given in a note to Art. 51 is a good 
exercise for students. 



THE CALCULUS 153 

98. When using squared paper we saw that many other things 
are summed up exactly as areas are calculated. Thus, if y is the 
area of cross-section of a body with a straight axis at the distance x 
from one end ; if there is a law connecting y and x ; the integral 
of y with regard to x is the volume. Also the integral of xy divided 
by the volume gives the x of the centre of gravity. 

Exercise. Suppose the rate of change of the velocity of a body 

dv 
per second -^, which we call the acceleration, to be constant, being 

a feet per second per second. The integral of this with regard to 
t is v^ the velocity, and it is 

v = at + Cj (1) 

where c is some constant. If v is Vq when ^ = 0, then c is Vq, But 

V is -J-, so integrating again, we have 
dt 

s^^-at^ + v^t + SQ, (2) 

for we add a constant on integrating, and we see that this constant 
must be 5q, the distance of the body from the zero of measurement 
when ^ = 0. 

Newton used the symbol s for velocity ; we may use this or 

^. He used *• or v for acceleration, which is our -j^ or ^. 
at dt^ at 

If we differentiate (2) we get (1); if we differentiate (1) we 
return to our old statement of acceleration = a. 



99. The following values of x and y being given, find Sy/8ic. 

Find also the values of y^x and add to get A^ the area of the 
«/-curve. 

The student must note that ^ is the average rate in each interval, 

ox 

he does not know the exact value of x for which this is the rate. 

If we wish to know -^ for any value of x from tabulated numbers 

dx 

yery exactly, we must use some such rule as is given in Art. 100. 

Note that in computing y .8x 1 use the average value of y in 
the interval, y . 8x is the area of a strip like EPP'E' of Fig. 25, 
and ly . dx means the sum of the areas of strips, the Greek letter 
s or 2 being commonly used to express " the sum of all such terms 



154 ELEMENTARY PRACTICAL MATHEMATICS 

as." When we find an area accurately and take an infinite number 
of infinitely narrow strips we replace 2 by the long English s or I 
It is most important to remember that when we express the integral 
of y we not only use the symbol | but also dx^ that is, we use \y. dx. 



X 


y 




y.8x 


A or -Zyhx 








1-736 


0-00868 


0-0000 


01 


0-1736 


1-684 


•0-2578 


0-00868 


0-2 


0-3420 


1-580 


0-04210 


0-03446 


0-3 


0-5000 


1-428 


0-05714 


0-07656 


0-4 


0-6428 


1-232 


07044 


0-13370 


0-5 


0-7660 


1-000 


0-08160 


0-20414 


0-6 


0-8660 


0-737 


0-09028 


28574 


0-7 


0-9397 


0-451 


0-09622 


0-37602 


0-8 


0-9848 






0-47224 



The symbol \y.dx means "the area of the curve whose ordinate 

is y from x = a to x==b." We already know how to find this. Or, 
what comes to the same thing, this symbol tells us, "Find the 
general integral of y, insert in it x = b, insert in it x = a, subtract." 
This is evident from Art. 97. 

The student is again warned that he must not be frightened by 
algebraic symbols. A symbol is the handiest way of telling you 
exactly what to do. It is usually much easier to understand than 
the hieroglyphics scratched by tramps on a farmer's gate to give 
information about savage dogs or charitable women. It is the 
easiest thing imaginable to get familiar with innocent symbols like 
these that I have been using ; yet the average student shuns them 
and never tries to understand them, because he has made up his 
mind that he cannot ever understand them. 

Exercise. The parabola y = 0-lxi revolves about the axis of x 
and generates a paraboloid of revolution. Let the student plot 



THE CALCULUS 



155 



the curve and study the shape of the surface Now the area of 
the cross-section at x is iry^ ; the volume of a slice between two 
cross-sections at the distance dx asunder is 

and the integral of this is the volume. Thus between sections 
dXx = a and x = h the volume is 



y^ . dx or 






dx or 



TT 

Too 



[H 



or 



IOOV2 2 



or 



200 



(62 - a^ 



If a = 0, so that we wish to know the volume from a; = to « = 6, 
the answer is -^V^. Call this V. 



200 

If instead of y = 0'\x^ we had y = mxi, then F would be \iim%'^. 
If yj is the ordinate of the curve where x is h, it is easy to show 
that V=\Try^h. But iry^ is the area of the end section, and we 
see that if a cylinder, a paraboloid, and a cone are on the same 
circular base and of the same vertical height their volumes are as 
I to i to 1. 

Illustrations. Let the student manufacture a few illustrations 
like the following : 



1. To illustrate that when 



sin a;, ^ = cosa;. 
dx 



Angle in 
degrees. 


X the angle 
in radians. 


y = sin X 


% 


Byl^x 


Average 
BylBx. 


cosx 


40 
41 
42 


0-6981 
0-7156 
0-7330 


0-6427876 
0-6560590 
0-6691306 


00132714 
0130716 


0-7583 
0-7512 


0-7547 


0-7547 



2. To illustrate that when y = cos x^ -t-= - sin x 



Angle in 


X the angle 




-Sy 


% 


Average 


degrees. 


in radians. 




hx 


Sy/Sx. 


20 


0-3491 


0-9396926 


00061122 


- 0-3513 




21 


0-3665 


0-9335804 


00063965 


-0-3656 


-0-3584 


-22 


0-3840 


0-9271839 









0-3584 



156 ELEMENTARY PRACTICAL MATHEMATICS 



3. To illustrate that when y = log ic, —- = -. 

ClX X 



X 


y = \ogx 


s. 


Sx 


1 

X 


213 


0-7561 


0-0047 


0-47 




214 


0-7608 


0-0047 


0-47 


0-47 


215 


0-7655 









This illustration would be more interesting if I used seven-figure 
instead of only four-figure logarithms. 

Remember that log x is the Napierian logarithm of x. 

4. To illustrate that when y = e"", -,- — e^ 



X 


.V=e^ 


Si 


5?/ 
Average -^ 


2-000 
2-001 
2 002 


7-3889 
7-3962 
7-4037 


7-3 
7-5 


7-4 



5. There is a hyperbola y- 



100 



X 


p 


8?/ 
Sx 


ij.Sx 


Area or 
l.y.Sx. 


10 


10 


-0-909 


9-546 





11 


9 091 


-0-758 


8-712 


9-546 


12 


8-333 


-0-641 


8-013 


18-258 


13 


7-692 


-0-549 


7.418 


26-271 


14 


7-143 


-0-476 


6-905 


33-689 


15 


6-667 


-0-417 


6-458 


40-594 


16 


6-250 


-0-368 


6-066 


47-052 


17 


5-882 


-0-326 


5-719 


53-118 


18 


5-556 


-0-293 


5-409 


58-837 


19 


5-263 


- 0-263 


5132 


64-246 


20 


5-000 






69-378 



find its area from the ordinate 

at a: =10 to any ordinate 
for greater values of x. 

Find also -/ at any point. 

Tabulate as follows. Notice 
that to get y. 8x for the 
interval 8x=l from 10 to 
11, 1 take the average y, 
which is 9-546. 

The true answers are 
^__100 _y 

dx ~ x^ «' 



and area =100 log^ y^ ; 

and it will be found that 
the tabulated answers are 
fairly correct. 



THE CALCULUS 



157 



EXERCISES. 

1. The following numbers give ^ feet, the distance of a sliding piece 
measured along its path from a certain point to the place where it is at 
the time t seconds. Find the velocity and acceleration at various times, 

and draw three curves showing how x^v ov -^ and ~ or the acceleration, 



depend upon t. 



dt 



dt 



t 


X 


V 


Acceleration. 





1000 






0-05 




17-36 




01 


2-736 




-5-2 


015 




16-84 




0-2 


4-420 




-10-4 


0-25 




15-80 




0-3 


6000 




-15-2 


0-35 




14-28 




0-4 


7-428 




-19-6 


0-45 




12-32 




0-5 


8-660 




-23-2 


0-55 




10-00 




0-6 


9-660 




-26-3 


0-65 




7-37 




0-7 


10-397 




-28-6 


0-75 




4-51 




0-8 


10-848 




-29-9 


0-85 




1-52 




0-9 


11-000 




-30-4 


0-95 




-1-52 




10 


10-848 




--29-9 


105 




-4-51 




11 


10-397 







If a slider weighs 161 lb., what is the force acting upon it when ^=0*5. 

1 ft! 

Ans. A retarding force of -^7^ x 23-2 or 116 lb. 

2. By tabulation, give approximately a table of values of y. 8.v and 
A= \y.dx, if the following values of x and y are given. Let A be 
when X is 0. Plot y and x on squared paper and plot also A and x. 



y 

y.5x 
A 



0- 



0-2 



0-3 



0-4 



0-5 



0-6 



1-5663 1-6774 1-8002 1-9391 | 21000 2-2918 25-281 
0-16219 0-17388 0*18697 0-20196 0-21959 0-24100 
i 0-16219 j 0-33607 I 0*52304 i 0*72500 j 0-94459 I 1 18559 



168 ELEMENTARY PRACTICAL MATHEMATICS 



3. The following numbers give v the speed of a train in miles per hour 
at the time t hours since leaving a terminus. In each interval of time, 
what is the distance v . 8t passed over by the train ? At each of the 
times tabulated, what is .^, the distance from the terminus ? 



t 


V 


v.Sf 


X 








048 





004 


2-4 


0142 


0-048 


0-08 


4-7 


0-238 


0-190 


0-12 


7-2 


0-336 


0-428 


0-16 


9-6 


432 


0-764 


0-20 


120 


0-526 


1-196 


0-24 


14-3 


624 


1-722 


0-28 


16-9 


0-716 


2-346 


0-32 


18-9 


0-792 


3-062 


0-36 


20-7 


0-858 


3-854 


0-40 


22-2 


0-912 


4-712 


0-44 


23-4 


0-954 


5-6-24 


0-48 


24-3 


0-984 


6-578 


0-52 


24-9 




7-562 



Thus, for example, at t = 0-12, x is 
0-428 mile. In the next interval of 
0-04 hour the train goes 0-336 mile, 
so at ^=0-16 it is at the distance 
0-428 + 0-336 or ^=0-764. 



4. A curve y = hx^'^ passes through the point (^=5, y = 4) ; find h. 

Ans. As 4 = 6x52°, 6 = 0-07155. 

Find the area of the curve between and the ordinate at ^ = 5. 

Ans. Chx^^ .dx = ~ [a^^f = 5-715. 
Jo 3 5 

5. A vessel is shaped like the frustum of a cone ; the circular base is 
10 inches diameter, the top is 5 inches diameter, the vertical height is 
8 inches. If x is the height of the surface of a liquid from the bottom, 
express the diameter of its surface in terms of x ; express A its area ; 
express V the volume of the liquid in cubic inches. 

Ans.d=m-^x', ^ = |(lOO-y^+?|x2); 

F= 78-54^ - 4-9085.^2 + 0-1023.^3. 

6. There is a curve y = ax'^. If ?/ = 2-34 when .r = 2, and if y = 20-62 
when .^^ = 5, find a and n. Ans. a = 0-4511, w = 2-3745. 

Let the curve rotate about the axis of x, forming a surface of revolution. 
Find the volume of the slice between sections at .r and x + 8x. What is 
the volume between the two sections at .^ = 2 and .r = 5 ? 

Ans. ira^x^'^.Bx and 1156. 



THE CALCULUS 



159 



7. The curve y = a + hx^ is such that if ?/ = l 61, .r=l, and if x=A^ 
y = 532 ; find a and h. 

The curve rotates about the axis of x ; find the volume enclosed by the 
surface of revolution between the two sections at ^- = 1 and a- = 4. 

Am. a = l-08, 6=053, volume 11 r85. 

8. If y=0'l^, we know that -r-=0'2x and A= \y. dx = ^r^x^, taking 
^=0 when .v — O. Tabulate the following values of y ; tabulate -^ and 

y . 8x and A approximately. Show in three curves how v, -^, and A 

ax 

depend upon x. Test your numbers against the correct formulae. 



X 


y 


fix 


Mean values 
oiy. 


y.ix 


A 


True A. 


-0-2 


0-004 


-0-03 










-0-1 


0-001 


-001 













9-0 


0-01 


0-0005 


0-00005 








01 


0-001 


003 


0-0025 


0-00025 


0-00005 


0-000033 


0-2 


0-004 


0-05 


0-0065 


0-00065 


0-00030 


0-000267 


0-3 


0:009 


0-07 


0-0125 


0-00125 


0-00095 


0-000900 


0-4 


0-016 


0-09 


0-0205 


0-00205 


0-00220 


0-00213 


0-5 


0-025 


0-11 


0305 


0-00305 


0-00425 


0-00417 


0-6 


036 


0-13 


0-0425 


0-00425 


000730 


00720 


0-7 


0-049 


0-15 


0-0565 


0-00565 


0-01155 


01143 


0-8 


064 


0-17 


0-0725 


0-00725 


0-01720 


0-017067 


0-9 


0-081 


0-19 


0905 


00905 


0-02445 


0-02430 


10 


0-100 


1 






003350 


03333 



The student will find his ^ to be the true -/, but there are slight 
0^ dx 

errors in his tabulative values oi A. 



9. Repeat the above tabulative process for y=.r^ from x = () to x=\. 

We know that -f^=Zx'^\ and A=-x*. There are slight errors in the 

dv 
tabulative values of -/ and A. 
dx 



160 ELEMENTARY PRACTICAL MATHEMATICS 

10. C is the current in amperes flowing in an electric circuit of 
resistance r ohms and inductance I henries when v is the voltage at the 
time t seconds. If r = l, ^=0'01, c is known for the following values of t : 



t 


c 


Se 
St 


V 
















9950 


100 


0-0001 


0-995 










9850 


100 


0-0002 


1-980 






0-0010 


9-516 










9010 


100 


0-0011 


10-417 










8900 


100 


0-001-2 


11-307 






0-0100 


63-211 










3660 


100 


0-0101 


63-577 










3630 


100 


0-0102 


63-940 







From the tabulated numbers And ^ , and calculate 



8f 
rc-\-l 



dc 
dt' 



We see that v is constant in this example. 
11. It is known (see Art. 117) that if 

then 



-j^=^m{pt-e\ where tan. = ^. 



.(1) 



.(2) 



Let a = 200, p = 307r, r=l, Z = 0-01. Calculate c for the following values 

of t, and tabulate. From the tabulated numbers find -y-, and calculate v 

dt 

from (1) of last exercise. Now find the true values of v from (2), and 

compare your answers. 



t 


c 


fie 
Bt 


V 


True V. 


0-0150 


88-96 


10800 


197-5 




0-0151 


90-04 


10800 


198-6 


197-8 


00152 


9112 









THE CALCULUS 



161 



12. A body of weight W lb. hangs from the end of a spiral spring 
whose stiffness is such that it extends h feet for a pulling force of 1 lb. ; 
it vibrates, and at the time t seconds the body is s feet from its mid 
position ; it is known that if ^ = 32'2, 



Wdh s ^ 

17:9 + 7 = or 

g dt^ h 



dt^^Wh^-^' 



The solution (see Art. 121) of this equation is 

8 = A^m(nt->re) if n^ is -^. 
Wh 



(2) 

rr n 

A and e may have any values. 

Let A=0"01, Fr=64'4, ^ = 1, e = ; calculate s for the following values 
of t. From the tabulated numbers find -^, and try if (1) is satisfied. 

Remember that the angle nt is in radians. If the sum of the terms 
in (1) is not zero, call it ' error.' 



t 


s 


&8 

"=81 


Sv cPs 


'Error.' 


0-070 
0071 
0-072 


0-47500939 
0-48121983 
0-48740631 


6-21044 
6-18648 


-23-96 


+ 0-10 


0-140 
0141 
0142 


0-83599796 
0-83985741 
0-84367453 


3-85945 
3-81712 


- 42-33 


-0-34 



The student will find that even with seven-figure tables he does not 
get sufficient accuracy for the illustration of the true law. 

Exercises like this are troublesome, but they teach many useful lessons. 

100. To find -^ with greater accuracy from tables of y and x. 
dx 

Suppose that y has been tabulated for equidistant values of x. Let us 

take an example. 



X 


y 


sy 






90 


1463 


302 






95 


1765 


351 


49 


8 


100 


2116 


408 


57 


5 


105 


2524 


470 


62 


8 


110 


2994 


540 


70 


8 


115 


3534 


618 


78 




120 


4152 









We want to know -/ for 
dx 



"We tabulate the successive differences 

as shown 
.r=105. 

Now one-fifth of 408 is evidently too 
small and one-fifth of 470 is too great ; 
the average of these is not usually 
correct either. There is a rule, deduced 
by an application of Taylor's theorem, 
which I shall not here prove (it is 
proved in my book on Steam, page 240), 
which may be employed in such cases. 
Let h be the x difference (in this case 5). 
Note the figures in clarendon type. 



P.M. 



162 ELEMENTARY PRACTICAL MATHEMATICS 

|4{i(408 + 470)-l(70-67) + l(8 + 8)}. 

In this case 3^=87*59. 
dx 

In particular cases we can find -^ with great accuracy even from 

only two terms if we know a good empirical formula. For example, we 
know that with not very great but with some accuracy the pressure and 
temperature of steam are connected by the law 

Therefore we need to know h only. 

Suppose we know that when p = 2524, Q is 105, and when p = 2994, Q is 

110, and we wish to find -^ when Q is 105. 
du 

105 = a + 625240 2, 

110 = a + 629940 2, 

5 = 6(29940 2 -25240 2) = 0-16656 ^^ 6=30-003, 

^=0-2x30-003x2524-0 8 or ^=87-7. 
dp dd 



CHAPTER XXI. 

ILLUSTRATIONS. 

101. When we use the letters x and y, we are generally speaking 
of a curve. 

Then -^ is the slope of the curve at any place (that is, the 

tangent of the angle which the tangent to the curve makes with 

the axis of x) and the integral of y (denoted by the symbol ly. c?aj) 

is the area of the curve. But it is to be remembered that x and y 
may be any physical quantities, and other letters may be used. 
Thus, if s is the space passed through by a body in the time t, 

— is the velocity v of the body. Also -^ is called the acceleration 

of the body. Various symbols are in use : 
s and t for space and time. 

ds 
Velocity v or -77, but Newton used the symbol s. 

dv d^s 
Acceleration -n or -jr^. or Newton's s or Newton's v. 
at ar 

dh 
Rate of change of acceleration would be -n^. 

Note that -^ is one symbol ; it has nothing whatever to do with 
dv- ^2xs 

such an algebraic expression as -. — -^. The symbol is supposed 

merely to indicate that we have differentiated s twice with regard 
to the time. 

As an interesting example of using other letters than x and ?/, let 
us consider the kinetic energy stored up in a moving small body of 
mass m and velocity v which passes through the small space h in 



164 ELEMENTARY PRACTICAL MATHEMATICS 

the time 8t, gaining velocity 8v because the force F is acting upon it. 
The gain of energy by the body 8E is F. 8,% which is the work done 
by the force F acting through the space 8s. But F is m multiplied 

by the acceleration, or F=m— and 8s = v . 8t, so that 

8F = F.8s = m^v,8t = mv.8v, 
8t 

8E 

or -=r- = mv, 

bv 

But our equations are only entirely true when 8s, 8t, etc., are 
made smaller and smaller without limit ; hence 

dE 

dv ' 

or, in words, "the differential coefficient of E with regard to v is mv." 
If, then, we integrate with regard to v, 

E = - mv^ + c, 

where c is some constant. As there is no E when v is 0, c must 

be 0, and so we have ^ 

E^^mvK 

Practise differentiation and integration, using other letters than x 

and y. In this case -r- stands for our old -—- If we had had -~- = mx 
dv ax dx 

it might have been seen more easily that y = -^mx'^ + c. But you 
must escape from the swaddling bands of x and y. 



102. If the student knows anything about electricity, let him 
translate into ordinary language the improved Ohm's law 

,dc 
v = rc + lj^. 

Observe that r (ohms) and I (henrie.s) are known constants, so 

dc 
that if c and -j- are known variable functions of the time t, then 

V the voltage is known. The inductance / may be regarded as a 
back electromotive force in volts when the current increases at the 
rate of 1 ampere per second. 



ILLUSTRATIONS 165 

103. Ex. 1. Find the following integrals. The constants are 
not added in the answers : 



1 



v^.dv. Ans. -^v^. 



fv *dv. Ans. -^ v^-\ 
1-5 

I x/t^'-^ . f/«; or p^. f/v. Ans. -^v^. 

[r'^.dt. Ans. '2f. 
dv 



\ 



v + a 



. Ans. log(v + a). 



Ex. 2. It is proved in thermodynamics when ice and water or 
water and steam are together at the same temperature ; if ^2 is the 
volume in cubic feet of 1 lb. of stuff in the higher state, and if s^ is 
the volume of 1 lb. of stufl* in the lower state, then 

where t is the absolute temperature (being 273 + 6/° C), where I is 
the latent heat of one pound of stuff in foot pounds, j^ is the 
pressure in pounds per square foot. 

(i) In icewater,Si = 0-01 747, §2 = 0-0 1602 at ^ = 273 (corresponding 
to 0° C), p being 2116 lb. per square foot and / = 79 x 1400. Hence 

dv 79 X 1400 ^^^^^^^ 



dt 273(0-01602 -0-01747) 



Hence the temperature of melting ice is less as the pressure 
increases ; or pressure lowers the melting point of ice, that is, 
induces towards melting the ice. Observe the quantitative meaning 

of ^; the melting point lowers at the rate of 1 degree for an 
tit 

increased pressure of 279400 lb. per sq. foot, or 132 atmospheres. 

(ii) Water-steam. It seems almost impossible to measure accu 
rately by experiment s^^ the volume in cubic feet of 1 lb. of steam 
at any temperature, s^ for water is known. Calculate s^ - s^ from 
the above formula at a few temperatures, having from Kegnault's 
experiments the following table. The figures explain themselves. 
My calculation is for 105° C. 



166 ELEMENTARY PRACTICAL MATHEMATICS 







Pressure in 


St 


Assumed 


I 




9'C. 


absolute. 


lb. per sq. 
foot. 


dp 
dt 


foot- 
pounds. 


82 -«1 


100 


373 


2116-4 


81-5 








105 


378 


2524 


94 


87-8 


742500 


22-38 


110 


383 


2994 











It is here assumed that the value of ^ for 105° C. is half the 

dt 

sum of 81-5 and 94, because it is not worth while for my present 

purpose to find it more accurately by the methods of Art. 100 or 

by drawing a curve. 

Sg - Sj = 742500 -f (378 x 87-8) = 22-26. 

Now 5j for cold water is 0*016, and it is not worth while making 
any correction for warmth. Hence we may take 6*2 = 22-4, which 
is sufficiently near the correct answer for my present purpose. 

Ex. 3. The radius of curvature of any curve at the point (x, y) is 

(1) Find the radius of curvature of the parabola y = ax^ at the 
vertex, that is, when x = 0. 

Here -^ = 2ax. -j^ = 2a, and where x = 0, -/ = 0, r = ^ . 
ax ' aa;2 ' ^ dx la 

(2) Find the radius of curvature of the catenary 

1 . 



y = -ie''+e j. Ans. r 
At the vertex when x = 0, y = c, r = c. 



Ex. 4. The centre line of a uniform beam of length I fixed at one 
end loaded with the weight JF at the other, x being distance of a 
section from the fixed end ; it is known that y the deflection of the 
line from its unloaded straight form is 

y-Ei\:z 



l-> 



(1) What is the curvature where x = 01 
Here ~ = -p-J lx--x^ \ being where x = 0; 

d^y W „ . , . Wl . 



ILLUSTRATIONS 167 

Wl 
The curvature is therefore ^fy. 

1 Wl 

(2) What is the curvature where x=zil'i Ans. rrrrT' 

(3) What is the curvature where x = \. Ans. 0. 

In all practical cases in beams -^ is negligible in comparison with 
I, and I call it in (2) and (3). 



CHAPTER XXII. 
MAXIMA AND MINIMA. 

104. The student will see now that the problems of Art. 84 are 
easy to solve exactly. 

In (1). Divide a number n into two parts such that the product 
is a maximum. Let x be one part, then n-x is the other. Let the 
product be called «/, then 

y = x{n-x) or y = nx-x^. 

Now at a point of maximum, or minimum, the slope of the curve 

or -^ is 0. 
ax 

dv 
Here j- = ^^ - 2a;. If we put this equal to 0, 

?i-2a; = 0, 2« = ?i, x = -^. 

We get the required answer. 

It is true that we cannot in this way distinguish as to whether it 
is a maximum or a minimum that we have found, but there is no great 
difficulty in finding this out in other ways. I shall not give here 
the algebraic method of discriminating. 

In (2). When is the sum of a number and its reciprocal a 
minimum 1 Let x be the number; when is y = x-\-x~'^ a minimum? 

Ans. When j^ = or \-x~'^ = or a;-- 1=0 or x=\. 
ax 

In (3). The volume of the cylindric parcel is a maximum when 

^ is 0. Now V = 67r.'c2 - 27r%^ and ^ = 12'jrx- 6ir^xK Putting this 

equal to and dividing by Qttx, we get '2-irx = or x = -^ the 
answer obtained by the use of squared paper. 



MAXIMA AND MINIMA 



169 



In (4). Divide a number n into two parts, such that a times the 
square of one part plus h times the square of the other shall be a 
minimum. As before, 

y = ax^ + b(n-xY 

y = ax^ + biv^ - 2bnx + bx^ 

i/ = (a + b)x^ -\-bn^- '2bnx. 



or 
or 



Hence 



'^=2(a + b)x-2bn. 



Putting this equal to 0, we get 

x = J, the answer. 

a + b' 

In (5). The strength of a beam of rectangular section of given 
length, loaded and supported in any particular way, is proportional 
to the breadth of the section multi- 
plied by the square of the depth. If 
the diameter of a cylindric tree is a, 
what is the strongest rectangular 
beam which may be cut from the 
tree? 

Let x be the breadth BC of the 
rectangle BCEF (Fig. 27), BE being 
a. Then CE is Ja^ - xK Now the 
strength depends upon y = BCx CE'^, 
and this is 

y = x{a^ - x^) 
a maximum 1 




Fio. 2i 



or y = a-x 



x^. 



When is ^ 

-j- = o^ - 3a:2. Putting this equal to 0, we find 

3a;2 = a2 or x = -%- or 0-5774(fc. 



41 



In (6). y will be a maximum when its square is a maximum. 
The diameter of the tree being a, we desire that 
z = x^{oP--x'^) or ,^ = a-a^-a^ 

shall be a maximum. 

dz 



dx 



= 6aV - d>x\ 



170 ELEMENTARY PRACTICAL MATHEMATICS 

3 
Putting this equal to and dividing by x^, we find x^^-a^ or 

the depth x = -^a\/3 and the breadth = ^ a. 

These give the beam of greatest stiffness. 

Of course in all these cases differentiation gives us the correct 
answer, but our old squared-paper method has merits of its own. 

For example. Suppose there is a machine consisting of two parts 
whose weights are x and y. Now suppose the cost of the machine 
in pounds to be c^lOx + 3y. 

Also suppose the power of the machine to be proportional to xy. 
If the cost is fixed, find x and y, so that the power shall be a 
maximum. 



Here ^=o(c - lOx), 



and let 



p = xy or p 



XX:^{C 



10a;): 



10 



dp 
dx 



This 



20 . 1 

- -^x, and if we put this equal to 0, we have lOic = x ^• 

1 
leads to 3«/ = -c, that is, the costs of the two parts ought to be equal. 

Now suppose it is an actual machine, say a dynamo, and that x is 
the weight of the armature part, y the weight of the rest, and that 
the numbers 10 and 3 were based on real workshop experience. As 
everything is evidently proportional to c, take c = 3. Then 

10 . 



Now plot X and p on squared paper. It is quite true that we 
find a; = 0'15 gives the maximum power. But it will be observed 
that if this exactly best value of x is not employed 
the power is not much less than the maximum. 
In fact the squared paper tells us that we do not 
suffer greatly even by departing considerably from 
the value of x which is here thought to be the 
best. Indeed I may say that I employ the dif- 
ferential calculus method to suggest the best 
answer ; but I also use the squared paper method 
in practical problems. 



X 


P 


01 


0667 


012 


0-072 


013 


0-07367 


0-14 


007467 


0-15 


0-075 


016 


0-07467 


017 


07367 


0-18 


0-072 



MAXIMA AND MINIMA 171 

EXERCISES. 

1. When is ax — hx^ a maximum ? Ans. When a* = a/26. 

2. When is ax-hcfi a maximum ? Aiu. When x=slal^h. 

3. The volume of a circular cylindric cistern being given (no cover). 
When is its surface a minimum ? 

Let X be its radius and y its length, the volume is 

Trx'^y=a, say (1) 

The surface is Trx^+'2.Trxy (2) 

From (1), y is — - ; using this in (2) we see that we must make 

TTX^ 



7r^2 + 


— a minimum, 


2a 


=0 


or 


^ = 


= a/7r, 


x^= 


irxhf 

TT 


or 


x- 


=y- 



That is, the radius of the base is equal to the height of the cistern. 

4. Let the cistern of the last exercise be closed top and bottom ; find 
the condition that it shall have minimum surface with given volume. 
The answer is, that the diameter of the cistern is equal to its height. 

o. When a vessel moves at v knots, the total cost in wages, deprecia- 
tion, interest on capital, stores, coals, etc., is in pounds a-\-bi^ per hour. 
For a passage of m miles, what is the speed which will cause the total 
cost to be a minimum 1 The number of hours is mjv, so that the total cost 
is m{a + bv^)lv ; therefore we try to make av'^ + bv'^ a minimum ; that is, 

- av-^+2bv=0. 

The best speed is therefore v= yfal2b. 

The student may take a = 4 and 6 = 0*001 as being true of a certain 
cargo steamer whose speed I have studied. 

6. The sum of the squares of two factors of w is a minimum ; find them. 



If X is one of them, n/x is the other, and i/=x^+—^ is to be a minimum. 

-^ = 2x 5", and this is when x^ = n^ or x = \Jn. 

ax x^ 

7. The weight of gas which will flow per second through an orifice 
from a vessel where it is at a pressure pi into an outer atmosphere of 
pressure jOq is proportional to 

x-'^l-x ^ , 
where x is Po/Pi and y is a known constant ; when is this a maximum ? 



i+> 



That is, when is ^^ — ^ ^ a maximum ? 

Difi'erentiating with regard to x and equating to 0, we find 

2 \7/^Y-i) 
.yTT/ 



..= (: 



172 ELEMENTARY PRACTICAL MATHEMATICS 

In the case of air y = l"41, and we find p^) = 0'o27p^ ; that is, there is a 
maximum quantity leaving the vessel per second when the outside 
pressure is a little greater than half the inside pressure. 

In the case of wet or dry steam we may assume that we are dealing 
with a perfect gas whose y = ri3. The answer in this case is pQ = 0'578pi. 

8. From a hypothetic steam-engine indicator diagram the work done 
per cubic foot of steam is 

w=l44:pi(l + log r) - 1 44p3r, 

where pi and p^ are the initial and back pressures of the steam ; r is the 

ratio of cut off (that is, cut off is at -th of the stroke). 

r 

The logarithm is Napierian. If p^ and p^ are given, find r so that w 

may be a maximum. 

-7- = 144^ -144^3. Putting this equal to 0, we find the best r to 

hepi/p^. 

Taking pi = lOO lb. per sq. inch, find the best r in the following cases : 

(a) If j03 = 2, condensing engine. The best r is 50; cutting off at 
5^(y of the stroke gives most indicated energy per cubic foot of steam. 

(6) If ^3=10 + 2, the 10 representing the friction of the engine. The 
best r is 100-^12 or 8^ ; that is, cutting off at ^^ of the stroke gives 
most shaft energy per cubic foot of steam. 

(c) If p3=17, non-condensing engine. The best r is 100-^17 ; that is, 
cutting off at the 0*17 of stroke gives maximum indicated energy per 
cubic foot of steam. 

{d) If jt)3 = 17 + 8, where 8 represents the friction of the engine. The 
best r is 100 -r 25 or 4 ; that is, cutting off at j of the stroke gives most 
shaft energy per cubic foot of steam. 

9. Taking the waste going on in an electric conductor as consisting of 

(1) the ohmic loss ; the value of Ch' watts, where r is the resistance of a 
mile of going and coming conductor, and C is the current in amperes ; 

(2) the loss due to interest and depreciation on the cost of the conductor. 
It is easy to show that the total loss of money per annum is proportional to 



C^r + ~-\rh, 



where a and h are constants depending upon the price of copper and cost 
of manufacture and laying of the cable, and also upon the value of 
electrical energy. We may take a as never less than 1 7 and never greater 
than 40. To make y a minimum, 

^ = (7'2-aV-=^ = or Cr=a, 
dr 

Thus if a is 25, **=7>' 

When the cross-section of copper is a square inches, r is 0"04/a nearly, 
so that 0-04a(7=25 or aC=625. 

This means that in the most economical case we have 625 amperes per 
sq. inch of copper. 



MAXIMA AND MINIMA 173 

10. Find what positive value of ^ makes 

4^ + 3.^2-168^ + 10 
a minimum. A7is. ^=3*5. 

11. For what value of .r is 3sin.r+2cos^ a maximum, using (1) the 
calculus, (2) squared paper, (3) a geometrical method. Am. 0*9826 or 56°"3. 



CHAPTER XXIII. 

CURVES. 

105. When the equation to a new curve is given, the practical 
man ought greatly to rely upon his power of plotting it upon 

squared paper. Very often if we find -^ or the slope, everywhere, 

it gives us a good deal of information. 

If we are told that x^, y^ is a point on the curve, and we are 
asked to find the equation to the tangent there, we have simply to 
find the straight line which has the same slope as the curve there 
and which passes through x^,y^. The normal is the straight line 
which passes through the point x^, y^, and whose slope is minus the 
reciprocal of the slope of the curve there. 

3 1 

Ex. 1. The point a; = 4, y = 3 is a point on the parabola y = -^x^* 

[Try if it really is a point on the curve.] Find the equation of the 
tangent there. 

The slope is -f = -x-^ x~^ or as ic = 4 there, the slope is - x ^ or ^. 

CIX 'Z Z ty 4 J o 

The tangent is then y = a + -^x. To find a we have ^ = 3 when « = 4, 

3 

as this point is on the tangent, or 3 = a + ^ x 4, so that a is 1 J and 

3 

the tangent is y = l^+-^x. 

Ex. 2. The point a; = 32, ^ = 3 is a point on the curve y = 2 + ^x^. 
Find the equation to the normal there. 

The slope of the curve there is -/ = --x~^ = ——. and the slope of 

ax 10 160 

the normal is minus the reciprocal of this or - 160. 

Hence the normal is y = a -IQOx. But it passes through the 
point a; = 32, y = 3, so that 3 =a - 160 x 32. 

Hence a = 5123, and the normal is y = 5123 - 160a;. 



CURVES 175 

Ex. 3. At what point in the curve y = aa;"'' is there the slope b1 
As -/ = - wa«~''~\ the point is such that its x satisfies - 7iaa;~""^ = b. 
Knowing its x, we know its y from the equation to the curve. 

106. It is easy to see and well to remember that if x-^, y^ is a 
point in a straight line, and if the slope of the line is 6, then the 
equation to the line most quickly written is 

x-x^ 
Hence the equation to the tangent to a curve at the point iCj, y^ 
on the curve is ^ 

- — ^ = the -f- at the point, 

and the equation to the normal is 

i = - the -r at the point. 

y-Vi dx ^ 

Ex. 1. Find the tangent and normal to the curve 7ry'' = a at the 
point iCj, y^ on the curve. 

Ans. The tangent \b — x-\- — y = m-\-n, and the normal is 

^(x-x^)-^{y-yi)=^0. 

Ex. 2. Find the tangent and normal to the parabola y'^ = ^ax at 
the point where x = a. Ans. y = a + x, y — 3a-x. 

Ex. 3. Find t\ie tangent to the curve y = a + bx + cx^ + ex^ at a 

point on the curve x^^ y^. Ans. ^—^ == 6 + 2cx^ + Sex-^-. 

107. P (Fig. 28) is a point on the curve BFC at which the 
tangent PA and the normal PD are drawn. OX and OY are the 

axes. OPi = x, RP = y, tan PAX = -^; the distance AE is called 
the subtangent ; prove that it is y -r- -i^. 

The distance RD is called the subnormal ; it is evidently y^. 



176 ELEMENTARY PRACTICAL MATHEMATICS 

Y 




EXERCISES. 

1. Find the length of the subtangent and subnormal of the parabola 
«/=m,r2. Here -f^ = 2wi^, so that 

su btangent = mx^ -r 2m.^ or - .r, 

subnormal = ?/ X 2w.r or liin^ofi. 

2. Find the length of the subtangent of y = mx". 

Here -J^ = mnx^~^, so that 

X 

subtangent = mx"^ -t- mnx'^~^ = -. 

3. For what curve is the subnormal constant in length ? That is, 
dy dx \ 



y 



, —a or ^- = - 
dx dy a ' 



The integral of y with regard to ^^^ is -y^^ so that ^ = ^3/^ + a constant 6, 

say ; and this is the equation to the curve, b having any value. It is 
evidently any one member of a family of parabolas. 



CHAPTER XXIV. 



ILLUSTRATIONS. 



108. (1) The chain of a suspension bridge supports a load by 
means of detached rods ; the loads are about equal and equally 
spaced. Suppose a chain to be continuously loaded, the load being 
w per unit of hmizontal length. Any very flat uniform chain or 
telegraph wire is nearly in this condition. What is the shape of 
the chain ? Let be the lowest point. OX is tangential to the chain 
and horizontal at 0. OY is vertical. Let P be any point in the 



Y 


X 


y 


.To 


^ 


K 


^ 


> 


1VX 


X 



Fin. 29. 

chain, its co-ordinates being x and y. Consider the equilibrium of 
the portion OP. OP is in equilibrium under the action of T^ the 
horizontal tensile force at 0, T the tangential force at P, and u-x. 
the resultant load upon OP acting vertically. We employ the laws 
of forces acting upon rigid bodies. A rigid body is a body which is 
acted on by forces and is no longer altering its shape. 
P.M. M 



178 ELEMENTARY PRACTICAL MATHEMATICS 



If we draw a triangle whose sides are parallel to these forces, 
these sides represent the forces to the same scale. If 6 is the 
inclination of T to the horizontal (see fig. 30). 



and 
But 



wx 



COS 6, 



= tan6>. 



dy 



tan 6 is -/, so that -~ is 



w 



Hence, integrating, y 
C 



dx 
1 w 
2 1! 



dx 
r^^ + constant. 



-X. 



(1) 

•(2) 
.(3) 




Fig, 30. 



Now we see that y is when x is 0, so that the constant is 0. 
Hence the equation to the curve is 

1 w „ 



2r, 



.(4) 



and it is a parabola. From these statements all sorts of calculations 
are easily made. 

(2) The work done when fluid of the volume v and pressure p 
expands from the volume v^ to the volume V2 ^^ 

w=\ p.dv (1) 

If we know p as a function of v, it is easy to calculate tv. 
{a) 'Letp = cv~'\ the general integral of this is :j— ^ — v^~". Follow- 
ing the instructions of Art. 99, the answer is evidentl}^ 



■W 



'). 



ILLUSTRATIONS 179 

(b) The method fails when ?/'=!; that is, when j^v = c. 

The general integral of cv~'^ is clogv, so that in this case the 

answer is c{\ogv^-\ogv^) or clog-^. ' 

(3) The curve y = ax" revolves about the axis of x ; find the volume 
enclosed by the surface of revolution between the cross-sections at 
a; = and x = h. 

Evidently the volume of a slice of thickness dx is iry^. dx. Our 

answer is r& ^n^ 

or ^^ W''^\ 
'2n+\ 



fc^Q^. dx 




(4) It can be proved that when a perfect gas (whose law is 

•pv = lit if p is pressure, v volume, t absolute temperature, and B a 

constant) changes its volume and pressure in any way, the rate of 

dfT 
reception of heat by it per unit change of volume or -j- is 



.n dH 1 f dp 1 



where \ is the ratio of the two important specific heats. I always 

express heat in work units. 

(a) When gas expands according to the law^^^" = c a constant, find 

dH J dH y -n 

Ans. -j- = ^ — ^p. 
dv dv 7 - 1 

(/;) When the gas expands adiabatically ( that is —j- = j, what 

is n ? Ans. n = y; that is, the adiabatic law for an expanding gas is 
pv^ = constant. 

y is 1*41 for air and 1*37 for the stuff inside a gas or oil engine 
cylinder. 

(c) What is -r- when n=\1 Ans. -i- =p. 

(d) Notice that when 7i is greater than y the stuff is having heat 
withdrawn from it. 

(5) On the indicator diagram of a gas engine, here are some 
readings of p, pressure, and v, volume. The rate of reception of 
heat (if the gases are supposed to be receiving heat from an outside 
source and not from their own chemical action) is 



180 ELEMENTARY PRACTICAL MATHEMATICS 

J 77" 

Find -T-, which I call h. Plot p and v, and also h and v. Perhaps 
dv 

you had better plot h to a much smaller scale than j). 



V 


P 


8v 


dp 


dv 


20 


84-5 












255 


523 


1644 


2-1 


110 












660 


1419 


4048 


2-2 


176 












390 


878 


2877 


2-3 


215 












160 


376 


1721 


2-4 


231 












40 


98 


1060 


2-5 


235 












-90 


-229 


232 


2-6 


226 












-130 


-345 


-94 


2-7 


213 












-110 


-303 


-31 


2-8 


202 












-100 


-285 


-23 


2-9 


192 












-90 


-266 


-8 


30 


183 












-80 


-244 


+ 16 


31 


175 












-80 


-252 


-31 


3-2 


1(57 












-80 


-260 


-80 


3-3 


159 












-70 


-235 


-42 


3-4 


152 












-60 


-207 


+ 4 


3-5 


146 












-60 


-213 


-32 


3-6 


140 









CHAPTER XXV. 

ILLUSTRATIONS. BEAMS AND STRUTS. 

109. The curvature of a circle is the reciprocal of its radius, 
and of any curve at any point it is the curvature of the circle 
which best agrees with the curve at that point. The curvature 
of a curve is also "the angular change (in radians) of the 
direction of the curve per unit length." Now draw a very 

flat curve, with very little slope (or y^j everywhere. Observe 

that the change in -^ in going from a point F to a point Q is 

du 
almost exactly equal to the change of angle [change of -^ is really 

a change in the tangent of an angle, but when the angle is very 
small, the angle, its sine, and its tangent are all equal]. Hence the 

increase in -^ from F to Q divided by the length of the curve PQ is 

the average curvature from F to Q, and as FQ is less and less we 
get more and more nearly the curvature at F. But the curve being 
very fiat, the length of the arc FQ is really Sx, and the change 

in i]L divided by hx as 6a: gets less and less is the rate of change 
ax 

of ^ with resrard to r/*, and the symbol for this is -y-^- Hence we 
dx dx^ 

take the curvature of a very flat curve such as the centre line of a 

beam or strut to be ±-~,. 
dx^ 

It is proved in books on mechanics that the curvature of a beam 
or strut at any section is the bending moment M, there, divided 
by EI, if E is Young's Modulus for the material and / is the 
moment of inertia of the section about its neutral axis. 



182 ELEMENTARY PRACTICAL MATHEMATICS 

Ex. 1. Uniform beam of length I fixed at one end, loaded with 
weight JV at the other. Let x be the distance of a section from 
the fixed end of the beam, then M is JF{1 - x) if / is the whole 

length of the beam. Our y| = ^ may be written -^ -i-| = 1 -x. 

Integrating, we have, as B and / are constants, 
EIdy_ 1 , 

From this we can calculate the slope everywhere, but to know c it 
is necessary to know the slope at some one place. Now ^ is at 
the fixed end, that is where x = 0, so that c = 0. Integrating again. 

To find C, we know that y is when x is 0, so that C 
isO. 

We have then the shape of the beam 



Ti&'^'-r'' " 



I 
I 
HA 
I 
I 



I 
JO 



We usually want to know y when x is /, and this is B 
called D, the deflection of the beam, 

SEI 

Ex. 2. A beam of length / loaded with IF at the Q 
middle and supported at the ends. Observe that if half of 
this loaded beam has a casting of cement made round it so 
that it is rigidly held, the other half is simply a beam of 
length JZ fixed at one end and loaded at the other with J/F, 
and, according to the above result, its deflection is 

Exercises on this formula were given in Ex. 14, Art. 
28 ; they may be referred to again. 

110. Euler's Theory of Struts. Consider a strut per- 
fectly prismatic, of homogeneous material, its own weight 
neglected, the resultant force F at each end passing through 
the centre of the section there. Let PQR (Eig. 31) show the 
centre line of the bent strut. Let AB = y be the deflection at A^ 
where OA = x. Let OP = OR = /. «/ is supposed everywhere to be 
small in comparison with the length 21 of the strut. Fy is the 

bending moment at B and -^ is the curvature there if E is Young's 



Fiti. 31. 



ILLUSTRATIONS. BEAMS AND STRUTS 183 

Modulus for the material and / is the least moment of inertia of the 
cross-section about a line through its centre. It is not difficult 
therefore to see that 

EI~ dx^ ^^ 

dx^ ^ EI 

We find that y = ^ costix (2) 

satisfies (1) whatever value a may have. When a: = we see that 
y = a, so that the meaning of a is known to us ; it is the deflection OQ 
of the strut in the middle. Again, when x = l, y = 0. 

Hence acos7i/ = (3) 

Now, how can this be true ? Either a = or the cosine is 0. But 
if heiding occurs so that a has some value, the cosiim must be 0. 
That is, the angle nl must be 

2 ^" Wf7^2 ""' ^=-W 
is the load which will produce bending. It will produce very little 
or very much bending equally well. 

In my theory of struts I have shown why it is that this theory 
does not agree with experiment; experimental struts are not the 
perfect prisms perfectly loaded which the Euler theory assumes. 



CHAPTER XXVL 



ILLUSTRATIONS. FLUID. 



111. Suppose a mass of fluid to rotate like a rigid body about an 
axis with the angular velocity a radians per second. Let 00 be 
the axis. Let P be a particle of fiuid weighing w lb. Let OF = x. 

__i^.-_i_- 1 1 o.. Make FB represent 

this to scale and let FS represent w the 
weight, to the same scale ; then the re- 
sultant force, represented by F2\ is easily 
found and the angle IIFT which FT makes 
with the horizontal. -Thus 



Centrifugal force is mass '^ multiplied by a^x. 



w 



or 



S 



td.\\EFT^iv^-ah: 

g 

being independent of w ; we can therefore 
apply our results to a heterogeneous fluid. 
Now, if y is the distance of the point F 
above some datum level, and we imagine 
a curve drawn through F to which FT 
(at F) is tangential, and if at every point 
of the curve its direction (or the direction 
of its tangent) represents the direction of 

the resultant force 
evidently -g~-a^x. 

Integrating this, we find 

y= - ^ log ic + constant (1) 

The constant depends upon the datum level from which y is 
measured. This curve is called a line of force. Its direction at 



Fig. 32. 



if such a curve were drawn, its slope -/ is 



ILLUSTRATIONS. FLUID 



185 



any place shows the direction of the total force there. It is a 
logarithmic curve. 

Level Surfaces. If there is a curve to which FT is a normal at 
the point P, it is evident that its slope is positive, and in fact 

dy _a^ 

dx~~g ' 



so that the curve is 



P 



jr- a;2 + constant, (2) 



the constant depending upon the datum level from which y is 
measured. This is a parabola, and if it revolves about the axis we 




Fig. 33. 



have a paraboloid of revolution. Any surface which is everywhere 
at right angles to the resultant force on the particle is called a level 
surface, and we see.that the level surfaces in this case are paraboloids 



186 ELEMENTARY PRACTICAL MATHEMATICS 

of revolution. They are sometimes called equi-potential surfaces. 
It is easy to prove that the pressure is constant everywhere in such 
a surface, and that it is a surface of equal density, so that if mercury, 
oil, water, and air are in a whirling vessel, their surfaces of separation 
are paraboloids of revolution. 

The student ought to draw one of the lines of force (taking, say, 
a = 8, ,^ = 32-2) and cut out a template of it in thin zinc, 00 being 
another edge. By sliding along 00 he can draw many lines of force. 
Now cut out a template for one of the parabolas (its constant may 
be taken as 0) and with it draw many level surfaces. The two sets 
of curves cut each other everywhere orthogonally. Fig. 33 shows 
the sort of result obtainable. 



112. Motion of Fluid. If AB is a stream tube, in the vertical 
plane of the paper, consider the forces acting on the fluid which is 

between the sections at P and Q^ of 
length 8s feet along the stream and 
cross-section a square feet, where a 
and ^s are in the limit supposed to 
be infinitely small. Let the pressure 
at P be ^ lb. per sq. foot, the velocity 
V feet per second, and let P be at the 
vertical height h above some datum 
level R. At Q let these quantities be 
p + Sjy, v-\-&v, h + 8h. Let the fluid 
weigh w lb. per cubic foot. Find 
the forces urging PQ along the stream, that is, forces parallel to the 
stream direction at PQ. 

pa acts on one end P in the direction of motion, and (j)-{-8p)a acts 
at Q, retarding the motion. The weight of the portion between 
P and Q is a .8s .w, and as if on an inclined plane its retarding 
component is 




Veight X h^^igh^ o^ Pl^^^^ 
° length of plane 



or a .8s .w 



8h 
8s' 



Hence we have altogether, accelerating the motion from P 
towards Q, 



pa- {p-\-8p)a-a. 
dv 



. w. 



8h 



But the mass is a . 8s . w/g and -,- is its acceleration, so we have 

at 



ILLUSTRATIONS. FLUID 187 

merely to put the force equal to ^' ^'^ . -^. We have then, 

dividing by a, ^ 

„ «, Ss.wdv 

-&p-w. m= -ji- 

9 ^dt 

Now, if U be the time taken by a particle in going from P to Q, 
we know that «^ = ^ with greater and greater accuracy as hs is 
shorter and shorter. Also the acceleration is more and more nearly 
W7. Hence, if Ss is very small, ^s- -n = jr^ = v .8v, so that we have 

~dv + '^ + dh = (1) 

This is the fundamental equation of fluid motion. Integrating, 

-^ +— + /i = constant (2) 

2g ) w ^ ^ 

I leave the sign of integration on the ^ because w may vary. 

In a liquid where w is constant and in gases when pressures vary 
only a little (as in ventilation problems), 

— + ^ + A - constant (3) 

'2g w 

Ex. 1. Find what (2) becomes in the case of a gas in which the 

adiabatic law is followed, that is, w = cpy. Neglect h. 

1 y2 1 

Ans, If i = 1 - _, we have ^r- 4- — />*= constant. 

Hence, if the gas flows from a vessel where v = 0, ^j =p^ to a place 
where j^ =Pq and v = Vq, we have 

1 -y 2 1 2<7 

Ex. 2. Particles of water in a basin, flowing very slowly towards 
a hole in the centre, move in nearly circular paths, so that the 
velocity v is inversely proportional to the distance x from the central 
axis. Take v = a/x, where a is some constant. Then (3) becomes 

a^ p 
h + ^ — A + - = a constant. 
2gx^ w 

Now at the surface of the water p is constant, being the pressure 
of the atmosphere, so that there 



h = c 



2gx^' 



188 ELEMENTARY PRACTICAL MATHEMATICS 

This gives us the shape of the curved surface. Assume c and a 
any values (for example take c=], a = 0'S, g = 3'2), and it is easy to 
calculate h for any value of x (say from a; = 0*05 to a: = 5), and so plot 
the curve. This curve rotated about the axis gives the shape of the 
surface, which is a surface df revolution, c-his evidently the depth 
of any point below the level of the water at a; = x- . 

Ex. 3. Water flowing spirally in a horizontal plane follows the 
law V = a/x if x is distance from a central point. Show that 

As an example, take c= 10000, to = 62-3, </ = 32'2, a = 70, and draw 
the curve showing how jj varies from x=l to a; = 2. 



CHAPTER XXVII. 
THE COMPOUND INTEREST LAW. 

llZ.liy^be'^' (1), f^ = abe'- (2), or ^ = ay (3) 

Here is a function of x whose rate of increase is proportional to 
itself. If we ever see the statement (3) we can say that it is equi- 
valent to the statement (1), where b is any constant whatsoever, an 
arbitrary constant, as we call it. 

There are a great many phenomena in nature which have this 
property. Lord Kelvin's way of putting it was : " They follow the 
compound interest law." It is evident that b is the value of y 
when X is 0. 

Ex. 1. An electric condenser of capacity k farads is discharging 
through a great resistance r ohms. If v is the potential difference 
in volts at any time t between the coatings of the condenser, the 
quantity of electricity in the condenser is Q = kv. The current from 

the condenser is v/r, and this is also - -^, being the rate of diminu- 

dv 
tion of Q per second, or - k -jj, so that 

jdv_v dv _ 1 
~''Tt~? di~~hr'"- 

This is of the form (3) given above, v being our old //, / being our 

old Xj and - r-. being our old a. Hence we can say that 

V = be~i^-r, 
so that log b - log v = ^. 

This knowledge gives us a means of measuring the leakage 
resistance of a cable or condenser. When a condenser is steadiTy 



190 ELEMENTARY PRACTICAL MATHEMATICS 

losing its charge, if we measure v^ at the time t^ and we measure v^ 
at the time L, i n i i \ 4 

kr(\ogb-\ogV2) = t.2. 
Subtracting, kr(\og v^ - log v^) = t^-t^, 

so that r=r:^-j~, ^ — '^ 



^(logVi-log«;2) 
These are Napierian logarithms. If we use common logarithms, 

r = 0-4343 (^2 -y/^logio^. 

Thus, suppose a condenser of 2 microfarads (or 2 x 10~^ farads) is 
discharging; we find «; to be 10 volts and 20 seconds afterwards v 
is 8*2 volts ; what is the leakage resistance ? 

Here /2-/i = 20, ^^ = 2x10-^, 

0-4343x20 ^^ , ^r.c y. 

2x10 6(log 10 - log 8-2) 

Ex. 2. Neiotoii's Law of Cooling. Imagine a body all at the 
temperature v (above the temperature of surrounding bodies) to lose 
heat at a rate which is proportional to v. Thus, let 

dv 

where / is time. Then v = he~'^ 

or log h - log V = at. 

Thus, let the temperature be Vj at the time t-^ and v^ at the 
time t.^; then log v^ - log v^ = «(/.,- ^i), so that a can be measured 
experimentally as being equal to 

Ex. 3. A rod (like a tapering winding rope or like a pump rod 
of iron, but it may be like a tie rod made of stone to carry the 
weight of a lamp in a church) tapers gradually because of its own 
weight, so that it may have everywhere in it exactly the same 
tensile stress / lb. per sq, inch. If y is the area of the cross-section 
at the vertical distance x from its lower end, and if y + 5/y is its 
cross-section at the distance x + ^x from its lower end, then /. d>y is 
evidently equal to the weight of the little portion between x and 
iC + Sa;. This portion of volume is y. Sx, and if tv is the weight per 
unit volume, f.8y = w.y.h% or rather 

dy _w 

Hence, as before, y = he . 



THE COMPOUND INTEREST LAW 191 

If when x = 0, 2/ = yo ^^® cross-section just sufficient to support 
a weight J^F hung on at the bottom (evidently ft/^^ = TF), then ^^ = by 
because e^=l. 

It is, however, unnecessary to say more than that we have the 
law according to which the section of the rod alters. 

Ex. 4. Atmoqjhsric Pressure. At a place which is h feet above 
datum level, let the atmospheric pressure be p lb. per sq. foot ; at 
h + 5A, let the pressure be jf + Sp (pp is negative, as will be seen). 
The pressure at h is really greater than the pressure at h + 8h by the 
weight of the air filling the volume 8h cubic feet. If w is the 
weight in lb. of a cubic foot of air, - 8p = w. 8h. But w = q), where c 
is some constant if the temperature is constant. Hence -8p = c.p .8h, 
or rather dp 

i=-^ (1) 

Hence, as before, we have the compound interest law ; the rate of 
fall of pressure as we go up being proportional to the pressure itself. 
Hence p = be''''', where b is some constant. If p =Pq when h = (say 
at sea level), the law is p = v «"''^ 

As for c it is wJpq, Wq being the weight of a cubic foot of air at 
the pressure ^0- I^ ^ is the constant (absolute) temperature, and Wq 
is now the weight of a cubic foot of air at 0° C. or 273° absolute, 

then c is -^ . 

Po i 

But it is absurd to imagine that the temperature is constant. 
The following assumption is much more likely to be true. 

If w follows the adiabatic law, so that ^m'-v is constant (y is 1-4 
for air), then (1) becomes 

-8p = cp'^8h or -pj ^.dp = c.dh. 
Integrating, we get '—^ p y =ch + C. 

If p=Pq when A = 0, we can find 6', and we have (if ^-^ be 

called a), , y 

p)" =^o« - ach 

as the more usually correct law for pressure diminishing upwards 
in the atmosphere. 

Observe that when we have the adiabatic law and we have also 
p = Iitw, it follows that the absolute temperature is proportional 
to ])"; so that (3) becomes ah 

That is, the rate of diminution of temperature is constant per 
foot upwards in such a mass of air. 

Ex. 5. Compouml Interest. £100 lent at 3 per cent, per annum 
becomes £103 at the end of the year. The interest during the 



192 ELEMENTARY PRACTICAL MATHEMATICS 

second year being charged on the increased capital, the increase is 
greater the second year and is greater and greater every year. In 
fact, the increase of the principal every year is proportional to the 
amount of the principal. 

Here the addition is made every 1 2 months ; it might be made 
every 6 or 3 months or weekly or daily or every second. Nature's 
processes are, however, usually more continuous than even this. 

Let us imagine compound interest to be added on to the principal 
continually and not by jerks every year, at the rate of r per cent, 
per annum. Let F he the principal at the end of t years. Then 8F 
for the time 8t is 

r ffP r tL 

m^-'' "■• ^=100^' ''""«"'=« ^=^«^" 

In what time will a sum of money double itself at compound 
interest at r per cent, per annum 1 Here 

^ = log,2 or W = 69-31 or ^ = 69-31/7-. 

Students will find it interesting to refer to Arts. 26 and Ex. 3 of 
Art. 82. 

To what will the principal £100 amount in 20 years at 5 per cent, 
per annum 1 Calculate first by the above formula. Now calculate 
by the formula of Art. 26, the interest being added on yearly. 
Compare the answers. 

Answer by the above 271*8 pounds; the other answer is 265'3 
pounds. 

Ex. 6. Slipping of a belt on a pulley. When students make 
experiments on this slipping phenomenon, they ought to cause the 




Fio. 35. 



pulley to be fixed so that they may see the slipping when it 
occurs. 

The pull on the beU; at D is 2\, and this overcomes not only the 



THE COMPOUND INTEREST LAW 



193 



pull Tq Sit J, but also the friction between the belt and the pulley. 
Consider the tension T in the belt at B (Fig. 35), the angle 
^0^ being 6; also the tension T + 8T at C, the angle AOC being 

Fig. 36 shows BC greatly magnified, 86 being very small. In 
calculating the force pressing the small portion of belt BC against 



T+5T 



Fig. 36. 



the pulley rim, as we think of BC as a shorter and shorter length, 
we see that the resultant pressing force is T. 80* so that fiT. 86 is 
the friction if fx. is the coefficient of friction. It is this that 8T is 
required to overcome. When fx. T.86 is exactly equalled by 8T, 

dT 
sliding is about to begin. Then ^ . T. 86 = 82' or -rn^H-T, the com- 
pound interest law. Hence T= he^^. Insert now T=Tq when ^ = 0, 
and T= T^ when 6 = A0I)ot 6^, and we have 7\ = r^e'^^^ 

In calculating the horse-power H given by a belt to a pulley, M^e 
must remember that H=(T^- TQ)F-i- 33,000 if T^ and T^ are in 
pounds and V is the velocity of the belt in feet per minute. Again, 
whether a belt will or will not tear depends upon 1\ ; from these 
considerations we have the well-known rule for belting. (See 
Arts. 28 and 67.) 

For other illustrations of the Compound Interest law, the student 
may be referred to my book on the Calculus. 

Equation (3) is the simplest of a number of important statements 
which are referred to again in Arts. 134. 

* When two equal forces T make a small angle 5^ with one another, find 
their equilibrant or resultant. The three forces are parallel to the sides of an 
isosceles triangle like Fig. 37, where i^/> = Z)^ represents T, where FDE=8d 




Fig. 37. 



and EF represents the equilibrant. Now it is evident that as 56 is less and 
less, EF-rDE is more and more nearly 80, so that the equilibrant is more 
and more nearly T. 80. 

P.M. N 



194 ELEMENTARY PRACTICAL MATHEMATICS 

In the meantime, let the student consider the following two 
exercises : 

Ex. 7. Show that the equation -r^ - n^y = is satisfied by 

where A and B are arbitrary constants. 

Ex. 8. If i stands for V- 1, and if i behaves like an algebraic 
quantity so that *2= -1, i^= -i, i^ = l, i^ = i, etc., the equation 

■j^ + 72,2^ = is satisfied by 

It will be seen later on, that as M and iV may themselves involve 
the unreal i, this answer is equivalent to 

y = Asmnx-{-B cos nx, 

where A and B are arbitrary real constants. The student ought 
to try if this is the answer ; or what is equivalent, 

y = asm(nx + b), 

where a and b are arbitrary constants. 



GENERAL EXERCISES. 

The following exercises are on the subject matter of Chapters I. -XX VII, 
They are really questions which I have set in old examination papers. 

1. The value of a ruby is proportional to the 1"5*^ power of its weight. 
If one ruby is exactly of the same shape as another but of 2 "20 times its 
linear dimensions, of how many times the value is it ? A7is. 34-73 times. 

2. An annuity of £a per annum, the first payment being due one 
year from now, the last at the end of the n^^ year from now, has a 
present value • looj ( r \-^\ 

At 3 and at 5 per cent, per annum, find the present value of an annuity 
of £1 payable for 30 years. Ans. £19-57 at 3 %, £15-38 at 5 %. 

The present value of an annuity is £2000, it is to last for 20 years ; 
4 per cent, per annum. What is the annuity ? Atis. £148-8. 

The present value of an annuity of £65 is £630 at 4| per cent, 
per annum. What is the number of years' duration of the annuity ? 

Ans. 13 years. 

3. £100 being lent at compound interest at 4 per cent, for 20 years, 
find its amount — (1) If interest is added on yearly. (2) Every half-year. 
(3) Every month. (4) Continuously. 

Ans. (1) £219-11 ; (2) £220*8 ; (3) £222-25 ; (4) £222-56. 

4. An insurance office asks a present payment P for a life annuity 
of £a per annum (1st payment 1 year from now). If a person's 



THE COMPOUND INTEREST LAW 



195 



expectation of further life is n years and if the value of money is 
r per cent, per annum, 

If a is 1 and r is 3 %, find P in the following cases : 





Expectation 


Answers : 


Expectation 


Answers : 


Age. 


of life n. 


P 


of life n. 


P 




Men. 


Men. 


Women. 


Women. 


5 


50-87 


25-89 


53 08 


26-36 


15 


43-41 


24-05 


45-63 


24-65 


25 


35-68 


21-68 


37-98 


22-45 


35 


28-64 


19-0 


30-90 


19-92 


45 


22-07 


15-93 


24-06 


16-95 


55 


15-95 


12-5 


17-33 


13-33 



If a is 1 and P has the following values, find r. 



Age - - - - 


5 


25 


45 


MenP 


25-95 


21-6458 


16-4958 


Answers r • 


2-992 


3-025 


2-682 



A sum of money P being expended on a house, the lease of which 
has n years to run ; what is the equivalent addition a to the rent, 
r per cent, per annum being the value of money ? The above formula 
holds. 

In the following cases find a : 

(1) P = 1200, r = 5, n = %\. ^715. a = 93-57. 

(2) P=1500, r=4^, 7i = 20. Ans. a = 115-3. 
In the following cases find n : 

P=1200, r=5, a=90. Jws. w=22-5. 

To insure for a sum A to be paid at death, the letters r and n being as 
before, the premium being a, the first instalment payable at once. 

If a = l, r=3, find A in the following cases : 



Present age. 


Expectation of life. 
n for males. 


Answer A, 


5 
25 
45 


50-87 
35-68 
22-07 


112-57 
60-27 
29-90 



196 ELEMENTARY PRACTICAL MATHEMATICS 

In an insurance office if A = £100 and n has the following values, and 
also a, find r. 



Age. 


n 


a 


Answer r. 


21 
30 
40 


38-64 
32-10 
25-30 


21542 
2-5875 
3-3125 


1-013 

1-26 

1-54 



5. A sum of money A has to be provided for ; it is due n years 
hence. The annuity a is paid regularly to provide, the first payment 
being one year from now. 

If J = 100, r = 3-5 ; find a in the following eases : 



n 


5 


20 


40 


Answer a 


18-72 3-55 


1-189 



I suppose that all insurance offices are really borrowers of money, 
and therefore r is low. If they paid P now, to be repaid by yearly 
payments a for n years they would charge a large r. 



6. If 



A = 3P when r = 3i, 



and if 

find^i. Ans. 32-02. 

7. Two sides of a triangle are measured and found to be 32*5 and 
24-2 inches ; the included angle being 57°, find the area of the triangle. 
Prove the rule used by you. If the true lengths of the sides are really 
32-6 and 24-1, what is the percentage error in the answer ? 

Ans. 329-8, 0*12 per cent. 

8. In a triangle ABC, C being a right angle, AB is 1485 inches, AC in 
8*32 inches. Compute the angle A in degrees, using your tables. 

Ans. 55° 55'. 

9. ABC is a triangle. The angle A is 37°, the angle C is 90°, and the 
side ACia 5-32 inches ; find the other sides, the angle B, and the area of 
the triangle. Ans. 4009, 6-662, 53°, 10-66 sq. in. 

10. In a triangle ABC the angle C is 53°, the sides AC and AB are 
0'523 and 0*942 miles respectively ; find the aide CB in miles and the area 
of the triangle in square miles, either by actual construction on your 
paper or by calculation. Ans. 1-159 miles, 0-2420 square mile. 

11. In a triangle ABC, AD is the perpendicular on BC ; ABis 3-25 feet ; 
the angle B is 55°. Find the length of AD. If BC is 467 feet, what is 
the area of the triangle ? 

Find also BD and DC and AC. Your answers must be right to three 
significant figures. 

Ans. 2-66 ft., 6-21 sq. ft., DB=l-86, DC=2-8l, AC=3'81. 



THE COMPOUND INTEREST LAW 197 

12. A tube of copper (0-32 lb. per cubic inch) is 12 feet long and 
3 inches inside diameter ; it weighs 100 lb. ; find its outer diameter, 
and the area of its curved outer surface. 

Ans. 3*43 inches, 1552 square inches. 

13. The inside diameter of a hollow sphere of cast iron is the fraction 
0'57 of its outside diameter. Find these diameters if the weight is 60 lb. 
Take 1 cubic inch of cast iron as weighing 0"26 lb. 

If the outside diameter is made 1 per cent, smaller, the inside not being 
altered, what is the percentage diminution of weight ? 

Ans. 8-148 in., 4-644 in., 3-64 per cent. 

14. The cross-section of a ring is an ellipse whose principal diameters 
are 2 inches and 1| inches ; the middle of this section is at 3 inches from 
the axis of the ring ; what is the volume of the ring ? 

Prove the rule you use for finding the volume of any ring. 

Am. 44*43 cubic in. 

15. Let .V be multiplied by the square of y, and subtracted from the 
cube of z, the cube root of the whole is taken and is then squared. This 
is divided by the sum of ^, y, and z. Write all this down algebraically. 

Ans. ^^-^'f. 
x+y+z I 

16. Express ^~^^ 



as the sum of two simpler fractions. Ans. 



_2 1_ 

^+3 x—b 
What is the integral of the expression ? Ans. \og{x-\-2ifl{x — b). 



17. The sum of two numbers is 76, and their difference is equal to 
one-third of the greater ; find them. Ans. 45-6, 30-4. 

18. Suppose s the distance in feet passed through by a body in the 
time t seconds is s=10^''^. Find s when t is 2, find s when t is 2-01, and 
also when t is 2-001. What is the average speed in each of the two short 
intervals of time after ^ = 2 ? When the interval of time is made shorter 
and shorter, what does the average speed approximate to ? 

Ans. 40 ft., 40-401 ft., 40-04001 ft., 40-1, and 40-01 ft. per sec, 40 ft. 
per sec. 

19. It is known that the weight of coal in tons consumed per hour in 
a certain vessel is 0-3 + 0001 v^, where v is the speed in knots (or nautical 
miles per hour). For a voyage of 1000 nautical miles tabulate the time 
in hours and the total coal consumption for various values of v. If the 
wages, interest on cost of vessel, etc., are represented by the value of 
1 ton of coal per hour, tabulate for each value of v the total cost, stating 
it in the value of tons of coal, and plot on squared paper. About what 
value of V gives greatest economy ? Ans. i? = between 8^ and 8|. 

20. Write down algebraically : Add twice the square root of the cube 
of X to the product of y squared and the cube root of z. Divide by the 
sum of X and the square root of y. Add 4 and extract the square root of 

the whole. Ans. \ ^— - + 4h • 

y x-^ } 



198 ELEMENTARY PRACTICAL MATHEMATICS 

3^-2 



21. Express 



^2-3^-4 



2 1 
as the sum of two simpler fractions. Ans. 1 . 

22. Find two numbers such that if four times the first be added to 
two and a half times the second, the sum is 17*3 ; and if three times 
the second be subtracted from twice the first, the difference is 1 '2. 

Am. 3-229, 1-753. 

23. There are two quantities, a and h. The square of a is to be multi- 
plied by the sum of the squares of a and b ; add 3 ; extract the cube 
root ; divide by the product of a and the square root of h. AVrite down 

this algebraically. Ans. -^«H«^ + 6^) + 3, 

aslh 

24. Express -^ — ^ — as the sum of two simpler fractions and 

integrate. Ans. ^-^, logf^. 

25. A crew which can pull at the rate of six miles an hour finds that 
it takes twice as long to come up a river as to go down ; at what rate 
does the river flow ? Ans. 2 miles per hour. 

26. In any class of turbine if P is power of the waterfall and IT the 
height of the fall and n the rate of revolution, then it is known that for 
any particular class of turbines of all sizes 

n OC ^l-25p-05_ 

In the list of a particular maker I take a turbine at random for a fall 
of 6 feet, 100 horse-power, 50 revolutions per minute. By means of this, 
I find I can calculate n for all the other turbines of the list. Find n for 
a fall of 20 feet and 75 horse-power. Ans. 260. 

27. If JI is proportional to D^v^, and if I) is 1810 and v is 10 when ff 
is 620, find ff if D is 2100 and v is 13. Ans. 1503. 

28. liy=cuv^ + bxz'^, 

if y = 62*3 when ^=4 and 2=2, 

if y = 187-2 when ^ = 1 and 2= 1-46, 

find a and b, and find the value of y when .r is 9 and z is 0-5. 

Ans. a = 243-9, 6= -26-59, ^ = 671*9. 

29. If z=ax-bfx^. 

If 0=1-32 when x=l and 3/ = 2 ; and if z=8'68 when x=4: and y=l ; 
find a and b. 

Then find 2 when .37=2 and y = 0. Ans. a = 2*2, 6=0*11, 2 = 4*4. 

30. A cast-iron flywheel rim (0-26 lb. per cubic inch) weighs 13,700 lb. 
The rim is of rectangular section, thickness radially x, size the other way 
I'Qx. The inside radius of the rim is 14x Find the actual sizes. 

Ans. Thickness radially 7-124 in., other way 11-4 in., and inside 
radius 99-7 in. 

31. The electrical resistance of copper wire is proportional to its 
length divided by its cross-section. Show that the resistance of a pound 



THE COMPOUND INTEREST LAW 



199 



of wire of circular section all in one length is inversely proportional to 
the fourth power of the diameter of the wire. 

32. A hollow cylinder is 4*32 inches long ; its external and internal 
diameters are 3'150 and 1*724 inches ; find its volume and the sum of the 
areas of its two curved surfaces. 

Ans. 23*58 cubic inches, 66'16 square inches. 

33. A circular anchor ring has a volume 930 cubic inches and an area 
620 square inches ; find its dimensions. 

Ans. Radius of cross-section 3 in., mean radius of ring 5*2 in. 

34. The mean radius of a ring is 2 feet. The cross-section of the ring 
is an ellipse whose major and minor diameters are 0*8 and 0*5 feet ; what 
is its volume ? Ans. 3*948 cubic feet. 

35. The length of a plane closed curve is divided into 24 elements 
each of 1 inch long. The middles of successive elements are at the 
distances x from a line in the plane, as follows (in inches): 10, 105, 10*91, 
11-24, 11*49, 11*67, 12*57, 11*67, 11*49, 11*24, 10*91, 10*5, 10,10*5,10*91, 
11*24, 11*49, 11*67, 12*57, 11*67, 11*49, 11*24, 10*91, 10'5. 

If the curve rotates about the line as an axis describing a ring, find 
approximately the area of the ring. A ns. 1687 square inches. 

36. A prism has a cross-section of 50*32 square inches. There is a 
section making an angle of 20° with the cross-section ; what is its area ? 
Prove the rule that you use. An^. 53*56 square inches. 

37. Assuming the earth to be a sphere, if its circumference is 360 x 60 
nautical miles, what is the circumference of the parallel of latitude 56° ? 
What is the length there of a degree of latitude ? If a small map is to 
be drawn in this latitude, with north and south and east and west 
distances to the same scale, and if a degree of latitude (which is of course 
60 miles) is shown as 10 inches, what distance will represent a degree of 
longitude ? Ans. 12,080 miles, 33*55, 5*592 inches. 

38. The following table records the growth in stature of a girl A (born 
January, 1890) and a boy B (born May, 1894). Plot these records. 
Heights were measured at intervals of 4 months. 

Table of Heights in Inches. 



Year. 


1900. 


1901. 


1902. 


1903. 


Month. 


Sept. 


Jan. 


May. 


Sept. 


Jan. 


May. 


Sept. 


Jan. 


A 


54*75 


55*55 


56*6 


57*95 


59*2 


60*2 


60*9 


61-3 


B 


48*25 


49*0 


49*75 


50*6 


51*5 


52*3 


531 


53*9 



Find in inches per annum the average rates of growth of A and B 
during the whole period of tabulation. What will be. the probable 
heights of A and B at the end of another four months? Plot the rate 
of growth of A at all times throughout the period. At about what age 
was A growing most rapidly and what was her quickest rate of growth ? 

An^. 2*8, 2*4, 11^ years, 4*2 in. per annum. 



^00 ELEMENTARY PRACTICAL MATHEMATICS 

39. At a certain place where all the months of the year are assumed 
to be of the same length (30'44 days each), at the same time in each 
month the length of the day (interval from sunrise to sunset in hours) 
was measured, as in this table. 



Nov. 


Dec. 


Jan. 


Feb. 


Mar. 


April. 


May. 


June. 


July. 


8-35 


7-78 


8-35 


9-87 


12 


1411 


15-65 


16-22 


15-65 



What is the average increase of the length of the day (state in decimals 
of an hour per day) from the shortest day, which is 7*78 hours, to the 
longest, which is 16'22 hours? When is the increase of the day most 
rapid, and what is it ? Ans. 0-046 ; March, 0-072 hr. per day. 

40. Iipv''=a. 

If ^=100 when v=l, find «. Atis. 100. 

Taking pv^'= 100. Here are important exercises, in some of which n is 
0-8, another 0-9, and so on. Find in each case the value of p when v has 
the following values : 







Answers. 




rrivpTj VfllnPS 








of V. 


Value of p if 


Value of p if 


Value of p if 




71=0-9. 


n = l. 


n = l-13. 


1 


100 


100 


100 


1-5 


69-4. 


66-7 


63-2 


2 


53-6 


50 


45-7 


2-5 


43-8 


40 


35-5 


3 


37-2 


33-3 


28-9 


3-5 


32-4 


28-6 


24-3 


4 


28-7 


25 


20-9 



The student ought to show these answers on one sheet of squared paper 
as three curves. 



41. li pe=pi 



1 + loger 



■Ps- 



Let r = 3 and ^3 = 17. Find pe in the following cases 









Answers. 




Pl 


11 










Pe 


P 1 W 


w 


200 


2-273 


122-82 


140-8 


9241 


11320 


170 


2-649 


101-8 


116-7 


7929 


9716 


140 


3-177 


80-86 


92-7 


6610 


8100 


110 


3-984 


59-89 


68-66 


5272 


6462 


80 


5-370 


38-92 


44-64 


3911 


4793 


50 


8-340 


17-95 


20-59 


2519 


3085 



p= 



ApgRn 
33000' 



THE COMPOUND INTEREST LAW 201 

If >4 = 210, ^ = 1'2, 71 = 150, calculate P for each of the given values of p^. 

A ARn 

If v=-^^^^^x 60, show that V is 21000. 
144 r 

If IF is Vju, where each value of u is tabulated above, find W in 
each case. 

Plot the values of P and of W on squared paper, and try if there is a 
rule connecting them like W=^a4-hP 

and find a and h. Ans. a = 1400, 6 = 56. 

42. If TF' = (l+y)]f and if y=loli£, find W in each case of the last 

yjA7l 
exercise. Now plot W and P on the same sheet of squared paper as 
before, and see if there is a law connecting them like 

P=0-0144F'-24. 

In Exercises 41 and 42, p^ and p^ are the initial and back pressures of 
steam in a cylinder, pe the effective pressure ; cut off" at 1/r*^ of the stroke ; 
P the horse-power ; W lb. the indicated steam per hour ; W the actual 
steam per hour. 

43. ^=tan ^^tan (O + cfi), where <^ is always 10°, find ^ when 6 has the 
values 30°, 40°, 50°, 60°, and plot the values of x and of on squared 
paper. About what value of 6 seems to give the largest value of ;r? 
Ans. 40°. 

44. At speeds greater than the velocity of sound, the air resistance to 
the motion of a projectile of the usual shape, of weight w lb., diameter 
d inches, is such that when the speed diminishes from Vj feet per second 
to V, if t is the time in seconds and ^ is the horizontal space passed over 
in feet, 



^ = 7000 



w(l 



^• = 7000-^, 
d^ 



1} 



For a 12 lb. projectile of diameter 3 inches, if v^ is 2000, find x and t 
for the values of v, 1900, 1800, 1700, 1600, etc., down to 1000, and tabulate. 
For each of these values of t find 7/ = 2000at— 16'U^. Using x and y plot 
the curved path of a projectile, (P'^) when the initial inclination a is 0*12, 
(2°'^) when a is 0*06 radians to the horizontal. It is well to magnify the 
vertical dimensions.^ You have tabulated v, t, x and y. Now tabulate 
the values of x', which is xf = 2000^. Plot x' also with y ; this gives the 
parabolic path of the projectile if there is no air resistance. The eff'ect of 
air resistance ought to be carefully noticed in the two cases. 

45. One of my students in travelling to and from college sometimes 
noted times and distances on the railway. Here is a copy of one set 
of his observations. Find the average speed of his train in every 
interval. 

* The muzzle velocities in the two cases are 2014-4 and 2003 '6 ; the horizontal 
component of the muzzle velocity is 2000 in both cases. 



202 ELEMENTARY PRACTICAL MATHEMATICS 



Distances. 


Time 




Miles 
-Ss. 


Sec. 
St. 


Miles per hour 
Ss 














St 


Gravesend 1 
Central / 


Miles. 
- 22-21 

22 

21^ 
20i 


hrs. min, 

6 59 

7 00 

1 
3 


sec. 
24 

24 

45 

41 


•21 

•50 

1-00 

•50 


60 

81 

116 

52 


12-6 
22-2 
31 
34-0 


Northfleet 


- 20 
19i 


4 
5 


33 
19 


•50 


46 


39^1 










•50 


48 


37-5 




19 


6 


07 


•50 


40 


45 




m 


6 


47 


•50 


36 


50 


Greenhithe 


- 18 

m 

17 
16^ 
16 
n5-3 arr. 


7 
8 
8 
9 
9 
11 


23 
15 
34 
15 
59 
12 


'75 
•25 
•50 
•50 
•70 


52 
19 
41 
44 
73 


51^9 
47-3 
43-9 
40-9 
34-5 


Dartford - 


il5-3 dep. 
16i 
15 
141 

m 

14 
13 


12 
13 
14 
15 
15 
16 
17 


55 
37 
26 
02 
31 
20 
43 


•00 
•05 
•25 
•25 
•25 
•50 

1^0 

1^0 


103 
42 
49 
36 

29 
49 
83 
86 


00^0 
4-3 
18-4 
25 
31 
36-8 
43 •S 
41-8 


Bexley 


- 12 
11 


19 
20 


09 
46 


1-0 
10 


97 
94 


37-1 
38^3 


Sidcup 


- 10 
9 
8 
7 
64 
51 
5-6 


22 
23 
25 
26 
26 
29 
30 


20 
44 
00 
10 
48 
17 
06 


1^0 

1^0 

10 
•50 
•75 
•15 


84 
76 
70 
38 
149 
49 


42-9 

47-4 
51-5 
47-4 
18-1 
11-0 



THE COMPOUND INTEREST LAW 



203 



46. A summer student was given the dimensions, etc., of a mechanism. 
He made a skeleton drawing and found the following positions of a 
sliding piece (distance in feet from a point in its straight path) at the 
time t seconds. The periodic time is 2*4 seconds. Find the velocity 
and acceleration. 





X feet, the distance of 


%-^ 


Sv 


Time t seconds. 


slider from a point in 


r: or a 




its straight path. 


St 


St 





5-96 


-10-7 




0-1 


4-89 


-10-9 


-2 


0-2 


3-80 


-9-5 


14 


0-3 


2-85 


-7-6 


19 


0-4 


2-09 


-51 


25 


0-5 


1-58 


-3-2 


19 


0-6 


1-26 


-0-8 


24 


0-7 


118 


1-2 


20 


0-8 


1-30 


1-6 


4 


0-9 


1-46 


4-6 


30 


10 


1-92 


5-2 


6 


1-1 


2-44 


6-3 


11 


1-2 


3 07 


7-2 


9 


1-3 


3-79 


8-4 


12 


1-4 


4-63 


8-6 


2 


1-5 


5-49 


8-5 


-1 


1-6 


6-34 


7-8 


-7 


1-7 


7-12 


6-3 


-15 


1-8 


7-75 


41 


-22 


1-9 


8-16 


1-7 


-24 


20 


8-33 


-1-7 


-34 


2-1 


816 


-4-8 


-31 


2-2 


7-68 


-7-2 


-24 


23 


6-96 


-10-0 


-28 


2-4 


5-96 







Plot V and also a with x as abscissa and draw curves. Do the same 
with t as abscissa. 



204 ELEMENTARY PRACTICAL MATHEMATICS 



47. The present value of a lease which would bring in a net yearly 
value or annuity of £100 is as follows, interest being calculated at 
4 per cent, per annum : 



Number of years to run 5 10 


15 


20 


25 


30 


Present value 


445 


811 


1112 


1359 


1562 


1729 



Plot on squared paper. Find the present value if the number of years 
is 13. Ans. £992. 

48. The present price of an annuity of £100 to be paid to a person of a 
certain age is quoted from a certain insurance office advertisement. 



Age of annuitant - 


20 


30 


40 


50 


60 


70 


Price to be paid now 


2279 


2045 


1789 


1500 


1148 


797 



Plot on squared paper, and find the price to be paid for a £100 annuity 
for a person who is now 55 years old. Ans. £1334. 

49. By extracting square roots, we find successively 10^'\ lO*''^^ lO'''^^, 
Xooo625^ 10"03i25^ and 10° "i^e^s. By multiplying such results together we can 
find the following numbers : 



Numbers. 


Logarithms to the base 10. 


5-8294 
6-0429 
6-2643 


0-765625 
6-781250 
0-796875 



Plot on squared paper, and find the logarithms of 5-83, 6-00, 6-25 
correct to four significant figures. 

50. An army of 5000 men costs a country £800,000 per annum to 
maintain it ; an army of 10,000 men costs £1,300,000 per annum to 
maintain it ; what is the annual cost of an army of 8000 men ? Take the 
simplest law which is consistent with the figures given. Use squared 
paper or not, as you please. Ans. £1-1 x 10^ 

51. An examiner has given marks to papers; the highest number of 
marks is 185, the lowest 42. He desires to change all his marks according 
to a linear law converting the highest number of marks into 250, and the 
lowest into 100 ; show how he may do this, and state the converted marks 
for papers already marked 60, 100, 150. Use squared paper or mere 
algebra, as you please. Ans. 118-9, 160-8, 213-3. 

52. At the following draughts in sea water a particular vessel has the 
following displacements : 



Draught h feet - 


15 


12 


9 


6-3 


Displacement 7^ tons - 


2098 


1512 


1018 


586 



What are the probable displacements when the draughts are 11 and 
13 feet respectively ? Ans. 1350, 1700. 



THE COMPOUND INTEREST LAW 



205 



53. Work the following three exercises as if in each case one were alone 
given, taking in each case the simplest supposition which your information 
permits : 

(a) The total yearly expense in keeping a school of 100 boys is £2100 ; 

what is the expense when the number of boys is 175 ? 

(b) The expense is £2100 for 100 boys, £3050 for 200 boys ; what is 
it for 175 boys? 

(c) The expense for three cases is known as follows : 

£2100 for 100 boys ; £2650 for 150 boys ; £3050 for 200 boys. 
What is the probable expense for 175 boys ? 
If you use a squared paper method, show all three solutions together. 

Ans. (a) £3675 ; (6) £2812-5 ; (c) £2860. 

54. The following are the areas of cross-section of a body at right 
angles to its straight axis : 



A in square inches - 


250 


292 


310 


273 


215 


180 


135 


120 


X inches from one end 


22 


41 


70 84 


102 


130 


145 



What is the whole volume from x=0 to ^=145 ? 

At ;r = 70, if a cross-sectional slice of small thickness 8x has the 

volume 8v, find ~^. Ans. 33,420 cubic inches ; 273 sq. inches. 

55. h is the height in feet of the atmospheric surface of the water in a 
reservoir above the lowest point of the Dottom ; A is the area of the 
surface in square feet. 

When the reservoir was filled to various heights the areas were 
measured and found to be : 



Values of h 





13 


23 


33 


47 


62 


78 


91 


104 


120 


Values of A 





21000 


27500 


33600 


39200 


44700 


50400 


54700 60800 69300 



How many cubic feet of water leave the reservoir when h alters from 
113 to 65 ? Ans. 2*635 x 10^ cubic feet. 

56. There is a curve whose shape may be drawn from the following 
values of x and i/ : 



X in feet - 


3 


3-5 


4-2 4-8 


y in inches 


10-1 


12-2 


131 11-9 



Imagine this curve to rotate about the axis of x describing a surface of 
revolution. What is the volume enclosed by this surface, and the two 
end sections where .r=3 and ^ = 4*8 ? Ans. 6 cubic feet. 

57. The New Zealand Pension law for a person who has already lived 
from the age of 40 to 65 in the colony is : 

If the private income / is not more than £34 a year, the pension P is 
£18 a year. If the private income is anything from £34 to £52, the 



206 ELEMENTARY PRACTICAL MATHEMATICS 

pension is such that the total income is just made up to £52. If the 
private income is £52 or more there is no pension. 

Show on squared paper, for any income / the value of P, and also the 
value of the total income. If a person's private income is say £50, how 
much of it has he an inducement to give away before he applies for a 
pension ? Show on the same paper the total income, if the pension were 
regulated according to the rule. 

P= 18 - ^/. Ans. Anything up to £16. 

58. l{x=j \-cy-r' 

h-y ^ h 

If y is positive and never greater than h. If 6 = 10 and c = \. Plot the 
curve connecting x and y. 

P*whena=5; 2°"^ when a = l ; S'*^ when a = 0-1 ; 4**^ when a = 0-01. 

Plot these curves on the same sheet of paper after tabulating values of 
y and x. 



59. 



If ^ + ^ = 
25^16 



Calculate y for the 


following values of 


^, and tabulate as here shown : 


Given 
values of x. 





0-5 
or 
-0-5 


1-0 
or 
-1-0 


1-5 
or 
-1-5 


2-0 
or 
-2-0 


2-5 
or 
-2-5 


3 

or 
-3 


3-5 
or 
-3-5 


4 

or 
-4 


4-5 
or 
-4-5 


5 

or 
-5 


Any 

number 

greater 

than 5 or 

less than 

-5. 


Answers, 
values of y. 


±4 


±3-98 


±3-92 


±3-816 


±3-665 


±3-464 


±3-2 


±2-856 


±2-4 


±1-743 





Imaginary. 



Plot X and y on squared paper. 

60. The following corresponding values of two quantities, which we 
may call x and y, were measured : 



X 


0-5 


1-7 


3 


4-7 


5-7 


7-1 


8-7 


9-9 


10-6 


11-8 


y 


148 


186 


265 


326 


388 


436 


529 


562 


611 


652 



It is known that there is a law like 

y = a-{-hx 
connecting these quantities, but the observed values are slightly wrong. 
Plot on squared paper ; find the most probable values of a and 6, and 
state the probable error in the measured value of y when ^^ = 8*7. 

Ans. a = 119, 6 = 45*7, 2*4 per cent. 

61. In a price list I find the following prices of a certain type of steam 
electric generator of different powers : 



Kilovratts K. 


Price P pounds. 


200 

600 

1,000 


2,800 

7,160 

11,520 



THE COMPOUND INTEREST LAW 



207 



According to what rule has this price list been made up ? What is the 
list price of a generator of 400 Kilowatts ? 

Ans. P = 10-9A'+620 ; when A = 400, P=£4980. 

62. The following quantities are thought to follow a law like 
pt;»= constant. Try if they do so ; find the most probable value of 7i : 



V 


1 


2 


3 


4 


5 


p 


205 


114 


80 


63 


52 



A71S. n=0'8e. 

63. The following table gives corresponding values of two quantities 
.r and y : 



y 


10-16 


12-26 


14-70 


20-80 


24-54 28-83 


X 


37-36 


31-34 


26-43 


19-08 


16-33 


14 04 



Try whether a: any y are connected by a law of the form yaf^=c, and if 
so, determine as nearly as you can the values of n and c. 

What is the value of a- when 3/ = 17'53 ? Ans. i/a;^^^=bOl, 22-4. 

64. It is thought that the following observed quantities, in which 
there are probably errors of observation, follow a law like 

7/ = ae'*. 

Test if this is so, and find the most probable values of a and h. 



2-30 



3-10 



4-00 



4-92 



5-91 



7-20 



33-0 



391 



50-3 



67-2 



8-56 



125-0 



Ans. y=\Se^^. 

65. At an electricity works, where new plant has been judiciously 
added, if W is the annual works cost in millions of pence, and T is the 
annual total cost, and U the number of millions of electrical units sold, 
the following results have been found : 



u 


w 


T 


0-3 


0-47 


0-78 


1-2 


1-03 


1-64 


2-3 


1-70 


2-73 


3-4 


2-32 


3-77 



Find approximately the rules connecting T and W with U. Also find 
the probable values of W and T when U becomes 5, if there is the same 
judicious management. 

Ans. 2^=0-95^7+0-525, fr= 0-6 £/"+ 0-28 ; when £^=5, 7^=5-25, PF=3-28. 



208 ELEMENTARY PRACTICAL MATHEMATICS 



66. There is a function 

?/ = 5 logio^+6 sin J(y^' + 0"084(^- 3-5)2. 
Find a much simpler function of x which does not differ from it in 
value more than 2 per cent, between ^ = 3 and .r = 6. Remember that the 
angle ^^x is in radians. Atis. ?/ = 1*22^ + 49. 

67. Find accurately to three significant figures a value of x to satisfy 
the equation 0-5^i5- 12 logio^ + 2 sin 2^^=0-921. 

Notice in sin '^x that the angle is in radians. Ans. 1*22. 

68. The population of a country was 4-35 x 10« in 1820, 7-5 x 10^ in 1860, 
11 "26 X 10^ in 1890. Test if the population follows the compound interest 
law of increase. What was the probable population in 1910 ? 

Am. 14-78x106. 

69. Find accurately to three significant figures a value of x which 
satisfies the equation 

2^2 _ 10 logio X - 3-25 = 0. Ans. 1 '645. 

70. If p^lpi be called y. 

Let ^3=20 ; find y for the following values of p^. In each case find 
a value of x which satisfies the equation 

^ = --0*231og,o^. 



Pi 


200 


150 


100 


70 


50 


Answers, y 


0-10 


01333 


0-20 


0-2837 


0-40 


Answers, x 


4-135 


3-763 


3172 


2-60 


2-110 



71. If the values of p^ are as here given. If ^3 = 10, find in each 
case a value of x which satisfies the equation 
P3^1 l-25logio^ 



Pi 


200 


150 


100 


70 50 


Answers, x 


7-46 


6-35 


5-02 


4 03 


3-24 



72. Find, correctly to three significant figures, a value of x which 
will satisfy this equation : 

9.^3 - 41^ H0-5e2*- 92=0. Ans. 2-35. 

73. If y = 1^2 _ 3^ 4. 2, show, by taking some values of x and calculating y 
and plotting on squared paper, the nature of the relationship between 
X and y. For what values of ;r is y = ? Ans. ;r= 5-236, 0-764. 

74. The following values of p and u have been observed. It is thought 
that there is a law connecting p and u like 



THE COMPOUND INTEREST LAW 209 

try if this is so, and if it is nearly true, find the most fitting values of 
a and h. 



p 


1016 


20-80 


60-4 


101-9 


163-3 


225-9 


305-5 


u 


37-36 


19-08 


7-009 


4-29 


2-756 


2-031 


1-529 



Ans. pi*io«'»6^479. 

75. Mr. Odell made measurements of the torque c (in pound inches) 
required to keep a disc of paper of d inches diameter revolving at the 
following speeds n (in revolutions per minute), 

o?=22 inches. 



c 


0-33 


0-56 


0-875 


1-29 


1-76 


2-4 


n 


400 


500 


600 


700 


800 


900 



Plot log c and log n on squared paper, and see if there is a law, 

c cc 11^. Ans. m is 2-5. 

76. Do the same in the following case : 

Disc c?=27 inches diameter. 



c 


0-41 


0-575 


0-895 


1-297 


1-72 


2-2 


2-385 


n 250 

- 1 


300 


350 


400 


450 


500 


517 



Mean answer for both sets, wi = 2-5. 

77. The following measurements were made of the power P (in watts) 
required to keep a disc of paper of diameter d inches revolving at 
1000 revolutions per minute : 



d 


15 04 


21-82 


26-83 


36 


47-1 


P 


0-398 


3-162 


12-59 


50 


154-9 



Show that P X d\ and find s. Ans. s = 5'5. 

78. A firm is satisfied from its past experience and study that its 
expenditure per week in pounds is 

120 + 3-2^ + -^^ + 0-01(7, 
x + 5 ' 

where a; is the number of horses employed by the firm, and C is the 
usual turnover. 

If C is 2150 pounds; find for various values of a; what is the weekly 
expenditure, and plot on squared paper to find the number of horses 
which will cause the expenditure to be a minimum. Ans. 21 horses. 
P.M. 



210 ELEMENTARY PRACTICAL MATHEMATICS 

79. A number is added to 2*25 times its reciprocal ; for what number 
is this a minimum ? Use squared paper or the calculus as you please. 

A71S. 1'5. 

80. At the end of a time t seconds it is observed that a body has 
passed over a distance s feet reckoned from some starting point. If it is 
known that , = 25 + 150^-5^^ 

what is the velocity at the time t ? Ans. 150 — 10^. 

Prove the rule that you adopt to be correct. If corresponding values 
of s and t are plotted on squared paper, what indicates the velocity 
and why ? 

81. y = a-\-hx'^ is the equation to a curve which passes through these 
three points, 

^ = 0, y = l'24; ^=2-2, 3^ = 5-07 ; ^=3-5, y=12'64 ; 

find «, 6, and 7i. 

Find ^ when x = % Ans. a = VM, 6=0-602, 7i = 2-348, 3-593. 
ax 

82. The quantities V and C both vary with the time. If we know that 

dC 



V=RC+L 



dt' 



and that R = 0'\, L=000\ ; and if the values of C are as tabulated, find 
F approximately. Plot on squared paper. 



c 


t 


Calculated V. 


342-0 


0-000-20 


1675 


358-4 


0-00021 


1657 


374-6 


0-00022 


1648 


390-7 


0-00023 


1640 


406-7 


0-00024 


1632 


422-6 


0-000-25 


1623 


438-4 


0-00026 


1605 


454-0 


0-00027 


1596 


469-5 


0-000-28 


1578 


484-8 


000029 





83. There is a piece of a mechanism whose weight is 200 lbs. The 
following values of s in feet show the distance of its centre of gravity (as 
measured on a skeleton drawing) from some point in its straight path at 



THE COMPOUND INTEREST LAW 



211 



the time t seconds from some era of reckoning. Find its acceleration at 
the time ^ = 2*05, and the force in pounds which is giving this acceleration 
to it. 



s 


t 


0-3090 


2 


0-4931 


2-02 


0-6799 


2-04 


0-8701 


2 06 


1-0643 


2-08 


1-2631 


2-10 



Ans. 9-25 feet per sec. per sec. ; 57*5 lb. 

84. The following observed numbers are thought to follow a law like 
i/ = aa^/(l+sa;). Try by plotting the values of ^/x and 3/ on squared paper 
if this is so, and find the values of a and s. 



X 


0-5 


1 


2 


0-3 


1-4 


2-5 


y 


0-78 


0-97 


1-22 


6-55 


1-1 


1-24 



85. In the curve y=cx^, find c if y = m when x=h. Let this curve 
rotate about the axis of x ; find the volume enclosed by the surface of 
revolution between the two sections at x=a and x=h. Of course wi, 6, 

and a are given distances. Ans. — — ^-- K 

86. Find jp.dv, if pv*=c, a constant, 

(1) when s=0-8; (2) when s = l. 

Am. (1) 5cv°'2; (2) clog. v. 

87. If y=2'4-l-2x+0'2x^. find -^ and plot two curves from x=0 to 

dv ^^ 

x=4, showing how y and -^ depend upon .v. 

CCX 

88. Divide the number 20 into two parts, such that the square of one, 
together with three times the square of the other, shall be a minimum. 
Use any method you please. Atis. 15 and 5. 

89. If the current C amperes in a circuit is changing as shown in this 
table : 






279-3 


•296-7 


314-2 


331-6 


349-1 


366-5 


384-0 


t seconds 


0-1002 


0-1004 


0-1006 


0-1008 


01010 


0-1Q12 


0-1014 



and if V=RC+L^, where R is 0-3 and L is 8x10-4, find V. Plot C 

at 
and t^ and also V and t on the same sheet of paper. 



212 ELEMENTARY PRACTICAL MATHEMATICS 



t 


c 


dt 


V 


01003 


288-0 


69-6 


156-0 


01005 


305-45 


70-0 


161-63 


0-1007 


322-9 


69-6 


166-47 


0-1009 


340-35 


70-0 


172-10 


0-1011 


357-8 


69-6 


176-94 


0-1013 


375-25 


70-0 


182-60 



. 90. In Ex. 5, Art. 108, the values oi p and v on a gas-engine indicator 
diagram are given. The old assumption that y is a constant, is there 
made. But recent experiments have shown that we ought to take 

=2 + ^i7;, the stuff not being a perfect gas.* 

y — i oUO 

Repeat the work on this new assumption. I get the following answers : 



V 


2-05 


2-15 


2-25 


2-35 


2-45 


2-55 


2-65 


2-75 


h 


1750 


4868 


39-5 


2467 


1426 


353 


-273 


-103 


V 


2-85 


2-95 


3-05 


3-15 


3-25 


3-35 


3-45 


3-55 


h 


-144 


-112 


-69 


-137 


-202 


-140 


-66 


-115 



These results ought to be compared with those of Ex. 5, Art. 108, by 
both being plotted on the same sheet of paper. 

* ^=^ + T?u^ i^ ^ is ^^^ absolute temperature. In the above case t was 

'y — 1 1000 

563° when v was 2 and^ was 84-5, so that t = 3ipv. 



CHAPTER XXVIII. 
SIMPLE VIBRATION. 

114. The simplest periodic or vibratory motion is called Simple 
Harmonic Motion or s.h.m. In this case, if x is the displacement of 
a body from its mean position at the time t, 

X = a sin (qt + e) (1) 

In studying this let the student refer to Arts. 34 and 145, where 
I speak of the sine of any angle. He ought to watch the to and fro 
motion of the bob of a long pendulum, or the up and down motion 
of a cork floating on the sea when the sea waves have their simplest 
form, or the up and down motion of a weight hung from a spring 
balance. 

Ex. 1. Take a =10, 5^ = say ott ; take e = or ~ or any other 

value, and plot the curve. I often write 30° instead of - ; this is 

6 

incorrect, but convenient. It is seen that x cannot be greater 

than a and cannot be less than - a, and so a is called the amplitude 

of the motion, e is called the advance or lead or lag by various 

kinds of engineer. 

The angle qt + e is in radians. Now the sine of an angle is 

the same in every respect as the sine of the same angle plus 27r ; 

therefore, after the lapse of a time T (called the periodic time) the 

9_ 

values of x repeat themselves. That is, qT =^27r or q = ^ or 27r/ if / 
is - or the frequency or what musicians call the pitch. 

Ex. 2. A pendulum bob makes a total swing of 1"5 feet and 
executes the swing in 1-6 seconds; if the motion is given by 
(1), what are a and 3' 1 Ans. Evidently a = 0*75 foot. The periodic 

time T is that of two swings or 2'= 3*2 seconds; ^ = -—=1-96. 



214 ELEMENTARY PRACTICAL MATHEMATICS 

It is evident in (1) that x has the value a sine when ^ = 0. 
It is evident that a; = a sin {qt + 1 20) is the same as a; = a cos {qt + 30), 
so the curve plotted, although usually called a sine curve, may 
just as truthfully be called a cosine curve. 

The student will notice that the total area of the curve for a 
whole period is zero. He ought to spend much time in studying 
the motion, not merely getting the mathematician's knowledge 
of it, but making the sympathetic acquaintance with it which is 
necessary in the student of nature. 

Fig. 38 shows the shape of the curve. I advise the student to 
draw the curve by the following method. A little knowledge of 



9 2 ^^ 








S 


1 




















' 




.y^ >Ff^ — 


^ 












T x^ 


~ 


' 


" 




\ 






















/ 
















^ 










10 


■■ 






/ 




7 y V 


_ 


1- 


2- 


3- 


4- 


16 


i /-. 




" 










\ 














7 






M 


,V /1 3 
















\s 










/ 




\ /^ 


















^^^^C ~K^ 
















-A 


\ 






/ 




9 to '^ 






F^ 


1 






F— 1 


R 




v^ 


P^ 





Pig. 38. 



elementary trigonometry shows that it must give the same result as 
plotting. It is just what is done in drawing the elevation of a helical 
curve (as of a screw thread) in the drawing office. Draw a straight 
line OBM. Describe a circle with radius a. Set off the angle BOC 
equal to e. Divide the circumference of the circle from C into any 
convenient number of equal parts (16 in the figure), numbering the 
points of division 0, 1, 2, 3, etc. We may call the points 16, 17, 18, 
etc., or 32, 33, 34, etc., when we have gone once, twice or thrice 
or more times round. Take the distance BM to represent the 
periodic line of the motion and divide it into the same number of 
parts as you divided the circle. Number the points 0, 1, 2, 3, etc., 
16, B being and M being 16. Now project vertically and hori- 
zontally from corresponding points, and so get points on the curve. 
If OC is imagined to be a crank rotating uniformly against the 
hands of a watch in the vertical plane of the paper, x in (1) means 
the distance of C above OM, qt means the angle that the crank 
makes at any time with its initial position OCj q being the angular 
velocity of the crank in radians per second, and of course 27r/q means 
the time of one revolution of the crank or the periodic time T of 



SIMPLE VIBRATION 



215 



the motion ; x is the displacement at any instant, from its mid- 
position, of a slider worked by an infinitely long connecting rod. 

If a body has the motion (1), -7- is its velocity v at any instant, 

and -^ or -— is its acceleration. On the table of Art. 94, we 

see that if 



X = a sin (qt -\- e), 

dx 
V = -fj = aq cos (qt + e) = aq sin {qt-{-e-\- 90°), 



(1) 

(2) 



accel. a = ^- = -^ = -OAf- sin {qt + e) = a(f- sin {qt^-e-V 180°). ..(3) 

I give the proof of these formulae in Art. 95. But students 
ought to illustrate them in some such way as the following : 

Illustration. Let a = 1 0, q = 37r, e = - or 30°. Calculate x from ( 1 ) 

for the following values of t and tabulate. Now find _ in each 

interval, also find ~. Calculate aq cos (qt + e), which is the true 

value of V, and calculate - aq^ sin (qt + e), which is the true value of 
the acceleration. Your tables are not accurate enough ; you must 
work to six figures to get the acceleration approximately correct. 



t 


X 


fix 


Sv 


True V. 


True^;. 






fit 


St 




dt 


•060 


8-86204 


43 26 








•061 


8 •90530 


42-48 


780 


42^87 


791 


•062 


8 •94778 











You see that the true v is not very different from the mean of 
the two tabulated values 42-875, and the tabulated acceleration is 
not very wrong. 

Starting with (1), we might have written (2) as 

dx 

-n = (^ sin {qt + e + 90°), 

-^ = aq^sm(qt + e + l80°), 

and this fact cannot be too well remembered ; when we are dealing 
with such a function as (1), differentiation means multiplication 



216 ELEMENTARY PRACTICAL MATHEMATICS 

by q and increasing the lead by 90°. Integration means division 
by q and diminishing the lead by 90°. 

115. We see from Art. 114 that a function x = a sin qt is analogous 
to the straight-line motion of a slider driven by a crank of length a 
(rotating with the angular velocity q radians per second), by an 
infinitely long connecting rod. x is the distance of the slider from 
the middle of its path at the time t. At the zero of time x is ; 

q = 27r/= -„ , if T is the periodic time or if / is the frequency or 

number of revolutions of the crank per second. 

A function x = asin(qt + e) is just the same except that the crank 
is the angle e radians in advance of the former position ; that is, at 
time the slider is at the distance a sin e past its mid position. 

A function x = a sin{qt + e) + a' sin{qt + e') is the same as 
Z=^sin(^^ + ^); 
that is, the sum of two crank motions can be given by a single 
crank of proper length and proper advance. To prove this : 

Show on a drawing (Fig. 39) the positions of the first two when 
< = ; that is, set off YOP = e, 0P = a : YOQ = e\ OQ = a'. Complete 

X 




Fig. 39. 
[In every case we draw a crank in the position it has when ^=0.] 

the parallelogram OPRQ and draw the diagonal OR, then the single 
crank of length OR = A with the angle of advance YOR = E would 
give to the slider the sum of the motions which OP and OQ would 
separately give. Imagine the slider to have a vertical motion. 
Draw 0$, OR^ and OP in their relative positions at any time, ; these 



SIMPLE VIBRATION 



217 



relative positions do not alter, as the angular velocities are the same ; 
project P, B, and Q upon OX. The crank OF would cause the slider 
to be OF' above its mid position at this instant ; the crank OQ would 
cause the slider to be OQ' above its mid position ; the crank OR would 
cause the slider to be OB/ above its mid position at this instant; 
observe that OB' is always equal to the algebraic sum of OF' and OQ'. 

We may put it thus: "The s.h.m. which the crank OF would 
give, plus the s.h.m. which OQ would give, is equal to the s.h.m. 
which OB would give." Similarly, "The s.h.m. which OB would 
give, minus the s.h.m. which OF would give, is equal to the s.h.m. 
which OQ would give." We sometimes say : " The crank OB is the 
sum of the two cranks OF and OQ. In fact, cranks are added and 
subtracted just like vectors." 

These propositions are of great value when dealing with valve 
motions and other mechanisms. They are of so much importance 
to electrical engineers that many practical men who are fond of 
graphical methods of calculation say, "Let the crank OF represent 
the current." They mean, " There is a current which alters with 
time according to the law C = a sin{qt + e) ; its magnitude is analogous 
to the displacement of a slider worked vertically by the crank OF, 
whose length is a and whose angular velocity is q, and OF is its 
position when ^ = if the angle YOF is e." 

116. A simple case of the above is that 

a sin qt + b cos qt = a sin qt + b sin {qt + 90°) = A sin {qt + e), 
iiA^ = a'^ + b^ and if tan e = b/a. 

X 




This is easily proved trigono metrically. Graphically (Fig. 40), 
let OS=a, OQ = b. 



218 ELEMENTARY PRACTICAL MATHEMATICS 

The crank OP = A\& the vector sum of OS and OQ and tan e or 
taxiYOP = hla. 

117. I have spoken of vibratory motion and the motion of a 
slider, but it is to be remembered that our algebra applies to many 
other phenomena. 

The simplest alternating current of electricity follows the law 
c = CQsm{qt-\-e\ having a periodic time J which is equal to ^Trjq; it 
passes from a positive value c^ to the value -c^, and the way in 
which it varies with the time is shown on the curve (Fig. 38). 
e is called its lead, and depends upon what time we count from. As 
an illustration of the formula of Art. 116, let us pursue this 
electrical example very slowly and carefully. Why not take a 
whole week to it 1 

If the voltage in an electric circuit is v volts, the current c 
amperes, the resistance r ohms, and the inductance I henries, then, 
t being time in seconds, there is a well-known electrical law, 

^=^-^4^ w* 

The student ought to put this in non-algebraic language. 

dc 
Now if c = Cq sin qt, -j = c^q cos qt (see Art. 94), so that 

V = tCq sin qt + Ic^q cos qt. 
By the above rule, 

v = CQjW+¥^mi{qt + e) (2) 

where tan e = — . 

r 

If the amplitude of v is called v^, we see that VQ = CQ\/r^-\-Pq'^, and 
that V leads c by the angle e. Or again, we can say that 



' Jr^ + iy 

and G lags behind v by the angle e. 

In fact, if v = Vq sin qt, then c = , " sin {qt - e). The student 

must think this out clearly ; this statement is true if (2) is true. 

* We may suppose a circuit closed on itself and v is the electromotive force 
of an alternator in the circuit. In that case r and I will include the resistance 
and inductance of the alternator ; or we may suppose that we have only a 
branch connecting two points A and B, and v is the potential diflference 
established somehow between A and B. I often speak of a branch as "a 
circuit," because in each case there is no chance of a mistake being made. 



SIMPLE VIBRATION 219 

Remember that we can count from any zero of time, so long as 
we have the same zero of time when speaking of v and c. 
Jr^ + l^q^ is called the impedance of the circuit. 
It shortens our work often to write (1) in the symholic form 

v = ir-\-l-T\c, and we call t^ + l-r. an aperatoi' on c; or we write 

6 for J-. and say v = {r + l6)c or even that c = v-^{r + l6). The 
cct 

student ought to get accustomed to this symbolic language. 

118. Working graphically, let the crank OS (Fig. 40) represent 
re = tCq sin qt at the time t = and let all cranks be drawn in their 
position for ^ = 0. The length of OS is rc^. Then, as 

dc dc 

-J = G£ sin {qt + 90°), l-y is Ic^^q sin {qt + 90°), 

so let the crank OQ, set 90° in advance of OS, be of the length Iqc^. 
[In fact OS is r times the current and may be called the ohmic E.m.f. ; 
OQ is Iq times the current and may be called the reactance e.m.f.] 
The sum of these two cranks is OP, the total voltage. The length 
of OP is evidently 

slOS^ + OQ^ or c^slW+Pf and tan aS'C^P = /^/r = tan e. 
Again, it is well known (see Arts. 27 and 126) that if a circuit 
has not only ohmic resistance r and inductance I, but also a condenser 
of capacity k in series (see Fig. 43), the condenser sets up a 
negative reactance. In Fig. 41 
make 0*S'= rcg, make OQ = lqcQ, qT 

make 0K=^, the angles SOQ 

and /S'OiT being right angles and 
0^ just opposite to OQ ; that is, 
OK is a crank lagging 90° be- 
hind the current. The vector 
sum of OS, OQ, and OK is the 
total E.M.F. In fact, add the 
three cranks to get a crank p^^ ^^ 

representing the total E.M.F. 

In work of this kind we may use Cq and the other amplitudes, 
the length of crank in our answer being the amplitude of the 
voltage, or we can use the effective current instead of Cq, and 



220 ELEMENTARY PRACTICAL MATHEMATICS 



the length of crank in the answer will be the effective voltage. 
Effective current is O'TOTc^j and effective voltage is 0"707«?q.* 

119. Just as the rotating crank idea has produced an interesting 
graphical method of working exercises in alternating electric 
currents, so it has given great simplicity to algebraic calculation. 
In the above case, with no condenser, we have to add two vectors ; 
write them algebraically as OS = rcQ in the standard horizontal 
direction; Iqc^ in the direction OQ. Without Fig. 40 what are we 
to do '? We may use clarendon type, and say, 

OS + OQ = OP. 

This is my usual way of writing vector addition when I have no 
figure. And we may translate such a statement into 

* Current c is measured by an electric dynamometer, the torque or couple 
in which is proportional at any instant to the square of c. But c varies so 
rapidly that it is the average or mean value of c^ that is measured and the 
reading on the instrument is the square root of this, and this reading, what is 
called the effective current C, is "the square root of the mean square of c." 




Pig, 42. 

We graduate such an instrument using constant currents. If the current 
passes through a lamp of ohmic resistance r, the electric power converted into 
heat is Ch- watts. Now, if c = Cq sin qt, let the curve OAEBG in Fig. 42 
represent it for one period, AF being Cq. Let OA'EB'C represent the square 
of c. A'F is c^. It is evident that the average value of OA'EB'C (all whose 
ordinates are positive) is ^A'F or \c^, and the square root of this is c/\/2 
or '70700, and this is what we call G, the effective current. 



SIMPLE VIBRATION 221 

rc^ sin qt + Iqc^ cos qt = v 
or tCq sin qt 4- Iqc^ sin (qt + 90°) = v. 

Now, if instead of this we write 

{7' + Iqi) Cq sin ^^ = v, 
and we understand that i is an operator which advances a crank 
through 90°, it is more easily written and has many advantages if 
we can be sure that to give this meaning to i is perfectly consistent 
with algebraic truth. "A vector A multiplied by i turns it 
through 90° in the positive direction.""^ Again multiplying by i, 
we turn it through another 90°, or i^A means that A has been 
turned through 180°. Now ^^ jg _ i^ so that i^A is -A, and this 
algebra agrees with our notions of vectors. Again, ^^A is - ^A 
and z*A is A; that is, we have brought it to its original position. 
Throughout we see that our algebra agrees with* our notions of 
vectors. 

When the student has worked the exercises which will be given 
later, on the properties of i, which is \/- 1, and exercises on such 
expressions as a + hi and complex functions of such expressions, 
he will see the great importance of this way of thinking. 

We shall take real quantities and get real answers to problems, 
but we shall get these answers through the medium of unreal 
quantities. If the student is very mathematical, he will probably 
demand a proof that our methods of working are legitimate. I do 
not think that he would be satisfied with such a proof as I might 
give. But in any particular case it is always easy to prove that our 
answer is correct, and that there can only be one correct answer. 

In the above case, our symbolic operator o^ + l-j- or r + W has become 

d 
r + lqi by our assumption that qi may be used instead of ^ or ^.f 

We see that the operator a + hi when it multiplies or operates upon 
M sin qt multiplies Tli" by \faF+b^ and gives a lead e where tane = -. 

* The positive direction of angular increase is anti-clockwise. A Congress 
of Electricians last year determined to change this and to make the clockwise 
direction positive, but I adhere to the usual rule of all the mathematicians of 
the world. 

t Remember that it is only when we are dealing with simple periodic 
currents or functions of the time, of the type sin 5^ or 00s qt, that we replace 

^ or — by qi. 
dt ^ ^ 



222 ELEMENTARY PRACTICAL MATHEMATICS 



Similarly, when if sin ^^ is divided by a + bij it divides by s/a^ + b^ 
and gives a lag e where tan e = -. 

The operator q. evidently multiplies by Ja^ + b^, divides by 
v/a2 + ^2^ gives a lead tan"i- and a lag tan~i-- 

In fact, the result of multiplying M sin qt by -~ is 

-^VII?^'"(^'^'--'«-*^-"& (^> 

Until we deal with distributed capacity, it will be found that this 
formula will enable us to solve nearly every problem in alternating 
electric currents or the forced vibrations of mechanical systems. 

Ex. 1. A coil of r=l ohm and inductance / = 0004 is subjected 
to a voltage v = 141 -4 sin 600^. What is the current ? Here q = 600 
, . , 600 , , ^^ ,, 141-4 sin eOOif 

(a frequency of -^ or about 95 per second), ^^ i ^q.qq4 ^ 600i ' 

The denominator is 1 + 2 -41 Therefore the amplitude of c or Cq is 
141-4 ^^, 14r4 ^^ g^.gg ^^^ .J ^ .g ^^^ j^g^ tan e = 2-4 or 



x/1 + 2-42 2-6 

e = 67°-38. It would be more correct to give e in radians, but 
people have got accustomed to the use of degrees. 
The answer is then c = 54*38 sin(600i5 - 67°'38). 

The effective voltage is — —- or 0'707 x 141*4 or 100 volts. 

\/2 

The effective current is — — - or 0*707 x 54*38 or 38*45 amperes. 

Ex. 2. Part of a circuit has r = 100 ohms, inductance / = 0*09 henry 
in series with a condenser whose capacity is A; = 0-5xl0~^ farad 
[this is called a capacity of |^ a microfarad], and it has a potential 
difference or voltage established between its ends. [Or we may say, 
a whole circuit has the above r, /, and k, and the electromotive force 
in the circuit is ?;]. 'y = 14-14 sin 5000/, so that q = 5000 [a frequency 

of — — or nearly 800 per second]. What is the current? The 

1 i 

resistance is here r + Iqi + t-. or r + Iqi - ,— or 100 + i(450 - 400) 

orl00 + 50i. ^^^ ^^ 

14*14 sin 5000/ 14*14 • .^^^^, , 

c = — r^TTT — ^^. — = , sin (5000/ - e). 

100 + 50Z s/io¥TW 

'50 
c = 0*1264 sin(5000/-26°*57) as tan 6 = ^ = 0*5. 



SIMPLE VIBRATION 223 

The effective voltage is 10 volts, the effective current is 
0-707 X 0-2164 or 0894 ampere. The student ought to work 
this and the other exercises graphically also. 

Ex. 3. A circuit of r= 1000 ohms, no inductance, in series with 
a condenser of 0-5 x 10"'' farad has a voltage 14-14 sin 5000^. What 
is the current ■? Here the resistance is 

r-^= 1000 -400* 
kq 

J 14-14 sin 5000^ /%/moio • /K/^/^A^^ f^ioox 

^""^ '^ 1000-400i =Q-Q1313sm(5000^ + 2r-8). 

Notice how a condenser produces a lead just as an inductance 
produces a lag. 



CHAPTER XXIX. 
MAINLY ABOUT NATURAL VIBRATIONS. 

120. This chapter will seem to be difficult to a beginner, and 
some persons may think that it ought to be omitted in an elementary 
book, because in most practical problems, as the natural vibrations 
die out rapidly, we have only to deal with forced vibrations. But 
the man who studies it a little and, after working at forced vibrations, 
returns and makes a study of this chapter, will find that he is getting 
a very thorough knowledge of a subject exceedingly important in 
mechanics, electricity, acoustics, and light. 

I repeat that if a body has the motion (1)5-7- is its velocity at 
■.^ ^^ 

any instant and -j— is its acceleration ; we thus have 

Cit 

displacement x = asi\\{qt-\-e), ( 1 ) 

velocity v = -ri = aq cos {qt-\-e), (2) 

acceleration a = j- = -^ = - aq"^ sin (qt + e) (3) 

The most important peculiarity of this motion is that at every 
instant acceleration a oc displacement x 
or a= -q^x (4) 

or ^ + 4'^ = (5) 

If the equation (5) is given, we know that (1) is true, a and e 
being of any values whatever. 

121. Let us examine one case carefully. A weight of IF lb. 
hangs from a spring whose stiffness is such that a force of 1 lb. 
elongates it h feet. If the body is vibrating, when at the time t it 



MAINLY ABOUT NATURAL VIBRATIONS 225 

is X feet below (we imagine it moving downwards) its position of 

equilibrium, the force urging it to its position of equilibrium is 

x/h lb. ; this force is retarding the motion, and is equal to the mass 

cl'^x 
of the body Wjg multiplied by - -^72 • Now imagine that the 

motion of the body is also retarded by a force of friction which 
is proportional to the velocity, being h times the velocity let us 
say ; we have the total retarding force 

X vdx__}Fd^ 

h dt g dt^ 

W d^x ,dx x ^ ,,v 

or ' —-779+^:77 + 7 = (1) 

g dp' dt h ^ ' 

dH hq dx a ^ 

Let ^ be called 2/, let -^^ be called n^, and we have 

S + 2/S + »^. = (2) 

(1) or (2) expresses the damped motion of a vibrating body. I 
am neglecting the mass of the spring itself. 

It is shown in Art. 133 that the solution of (2) may have various 
forms, depending upon the amount of the friction. 

I. Assume no friction or h (or /) zero (2) now becomes 

and the complete solution of this is 

x = A sin nt + B cos nt, (3) 

where A and B are arbitrary constants. Of course (3) may be 

written in the shape . / , 

x = asm{nt + e), 

and A and B or a and e may be chosen arbitrarily or to suit any 

particular problem. For example, we may take the body to be at 

the extremity of its path or aj = a at time 0, and then e will be 90°. 

Or we may take the body to be at the middle of its path at time 0, 

so that e = 0, and if we state its velocity Vq at that instant, it is 

evident that a is Vq — n. As an example, take e = 90° ; this is the 

same as taking A = 0. In fact, let us take 

• x = B cos nt (4) 

P.M. p 



226 ELEMENTARY PRACTICAL MATHEMATICS 



II. When/ has a value but is less than n^ let s/n^ -/^ be called j?. 
The solution of (2) is (see Art. 133) 

x = e~^*{A mipt + Bco^pt) (5) 

Here again A and B are arbitrary constants. We can take them 
of any values we please to suit any particular problem. To compare 

with (4), let us take ^^Se-"co.pt (6) 

I will not now consider the case of / equal to or greater than ti, 
because cases of such excessive damping seldom come before us in 
practice, but all cases are easily solved. 

122. Let the student take up one case and consider it carefully. 

Q0.9 
Let ;F=64-4, A = 0-01, so that n^ = glWh = ^ ^ ^ ^ bO or 

w= 7-071. ^ 

Let ^=10, so that without friction 

a;=10cos 7-071/ (7) 

In (5) assume that /=0-5, p = slW^^ = JWlb = 1'0U, so that 

in this frictional case ,^ ..,, „^^., 

^=10e-"^'cos7-054/ (8) 

These two relations (7) and (8) ought to be plotted as curves on 

the same sheet of squared paper. The frictionless period T is — 

or 0-8886 second, and the friction increases it to — or 0*8907 second. 

p 

The amplitude of x diminishes as time goes on because of friction, 
so that at the end of a period it is only 0-64 of its value at the 
beginning of the period. (See Ex. 16, Art. 28.) 

Damped and undamped vibrations ought to be studied experi- 
mentally. 

W 
In the above formulae x may be angular displacement, — may be 

the moment of inertia of a body vibrating about an axis, and the 
stiifness of the restraining spring or torsional wire would be such 
that a unit torque or couple would produce a twist of h radians. 
A disc or wheel or rod surrounded by air, water, or oil, vibrating at 
the end of a twisting and untwisting wire is an excellent piece of 
apparatus to play with, and so create true instincts. 

The essential fact about simple vibration is that (neglecting 
friction) acceleration is proportional to displacement. Neglecting 



MAINLY ABOUT NATURAL VIBRATIONS 227 

the minus sign, since q is 27r/r, we see that to find the periodic time 
in any case ^^^^ / displacement ,9^ 

\ acceleration 

This may be linear displacement and linear acceleration or 
angular displacement and angular acceleration. Whether we study 
the simple pendulum or a compound pendulum or a vibrating 
column of liquid in a U-tube or the motion of the balance of a 
watch or the rolling of a ship, we readily calculate the periodic 
time by applying this rule. See my Applied Mechanics, Chap. XXV. 

Ex. 1. A body of 644 lb. has a simple periodic motion, the 
periodic time being 0*4 second ; the amplitude of the motion being 
1-5 feet (or total swing 3 feet). What forces are giving this motion 
to the body "? We can work with (9), and find if x is displacement 
and a is acceleration (without troubling about the minus sign) ; — 

a = -— — ic. The inertia or mass of the body is 644 -r 32*2 or 20, 

and force F in pounds is 20a. 

In fact F=4:935x. 

At the end of a swing F is 4935 x 1'5 = 7403 lb. At the middle 
of the swing the force is 0. The force is always in a direction 
towards the mid-point. 

Ex. 2. A body of 3*22 lb. is supported at the end of a strip of 
steel (whose inertia is neglected) ; the stiffness of the strip is such 
that a force of 1 lb. deflects the body 0*1 foot; the body is set 
swinging, what is its periodic time ? If a; is displacement in feet, 
the force in pounds acting on the body is F=10x; the inertia or 
mass of the body is 3*22 -=- 32*2 or 0*1 ; the acceleration is i^-r mass 
or lOOa^. Formula (9) gives 



2-= 2.^ 



j^ = f^ = 0-62832 second. 



Ex. 3. The steel rope to the cage of a mine moving downwards 
at 5 feet per second is suddenly stopped at its upper end ; neglecting 
the inertia of the rope itself, the cage which weighs 3220 lb. is now 
set in simple vibration, down and up. If the length and section of 
the rope are such that a stretch of 1 foot is produced by a force 
of 6000 lb., what is the greatest additional pull in the rope due to 
the stoppage ? What is the amplitude of the vibrational motion 'i 
What is the periodic time of the oscillation? The moment of 
stoppage is regarded by me as the zero of time. After this the 
cage has moved downward through x feet at the time t. I call x 
the displacement in the vibratory motion and take x = asmqt; 
V the velocity is aq cos qt and the acceleration a is -aq^ sin qt. 



228 ELEMENTARY PRACTICAL MATHEMATICS 

The inertia or mass is 3220 -i- 32*2 = 100; mass multiplied by 
acceleration is force, so that lOOa^^^gi-g^^gg^ force = 6000ft. Hence 
^2:^60 or q = 7-7iQ and ^=0-8112 second. The greatest velocity 
fl^ = 5, so that a = 0-6455 foot ; greatest force = 6000a = 3873 lb. 

123. Forced Vibrations. In the case considered (Art. 121), let 
the upper or supporting point of the spring be vibrated up and 
down. At the time t let it be lower than its mean position by the 
distance y. The spring is now elongated x-y more than if every- 
thing were at rest; the force retarding the body's motion is 

x-y jdx , , 
— r-^ + tj, and we nave 
h dt 



x-y ,dx 


WdH 




hg dx 


Wh^' 


= Wh« 



Therefore 5^+7^^+ Trjr^=Tri;?/ (1) 

or ^ + 2/J + »% = »2y (2)* 

Suppose now that y is a simple harmonic motion of any period, 
say that y = « sin g^ ; it is found that the body has a simple harmonic 
motion of this period, and along with this it probably has its 
own natural vibrating motion discussed in Arts. 120-122. But as 
there is always some friction, the natural vibratory motion is so 
rapidly damped out that it is of importance only for a short 
time — although it may be of considerable importance for that 
short time — and in much scientific work we neglect it. I shall 
therefore, after this, consider only the forced vibration. 

In most acoustic phenomena, although we assume that there is 
sufficient damping to destroy quickly the natural vibrations, we 
assume that / is because of ease in calculation and simplicity. 
This is so also in the study of light and some other sciences. But 
in the study of electro-magnetic radiation and alternating electric 
currents, the study of the tides and many other natural phenomena, 
we must take the / term into account. 

Exercise. If a body is hung from a spring, there being no friction, 
and if the support P gets a motion 

y = a^\w. qt^ 

*The interested student will consider another kind of forced vibration. 
Take y = 0, but assume that a downward force 7^= F^ sin qt acts upon the body. 
Instead of r^y in (2) we have i^hF. The results are much the same. 



MAINLY ABOUT NATURAL VIBRATIONS 229 



find the forced vibration. Here (2) becomes 



d^x 



+ ti^x = ii^a sin qt. 



(1) 



It will be found that x = A sin qt is the solution ; try it. 

dx d^x 

-^ = ^q cos qt, ^ = - ^(f sin qt ; 

and trying these values in (1), we find 

- Aq^ sin qt + ri^A sin qt = n^a sin qt. 

Therefore our guess is right if ^ = 



» 2 _ ^2 



It is well worth while to study this answer carefully, x is merely 



n- 



y multiplied by 

•ti>- — q- 

As n and q are proportional to the natural and forced frequencies, 
if the ratio of the forced to the natural frequency is called ^7, and if 
a = l, the amplitude of /F's motion being called A, we have 

1 



1-p 



2* 



p 


A 


p 


A 


•1 


1-01 


101 


-50 


•5 


r333 


103 


- 16^4 


•8 


2^778 


11 


-4 762 


•9 


5 •263 


15 


-0^8 


•95 


10 ^26 


2-0 


-0333 


•97 


1692 


5^0 


- 0^042 


•98 


25-25 


100 


-0 010 


•99 


50 25 






1 


00 







Note that when the forced frequency is a small fraction of the 
natural frequency, the forced vibration of JF is a faithful copy of 
the motion of the point of support P ; the spring and /F move like 
a rigid body. When the forced frequency is increased, the motion 
of TF is a faithful magnification of P's motion. As the forced gets 
nearly equal to the natural frequency, the motion of fF is an enor- 
mous magnification of P's motion. There is always some friction, 
and hence the amplitude of the vibration cannot become infinite. 
When the forced frequency is greater than the natural, /F is always 
a half period behind P, being at the top of its path when P is at the 



230 ELEMENTARY PRACTICAL MATHEMATICS 

bottom. This is the explanation of various interesting physical 
phenomena. For example, the dynamical theory of the tides of the 
ocean as differing from the older theory. 

When the forced is many times the natural frequency, the motion 
of W gets to be very small ; W is nearly at rest. 

Men who design earthquake recorders try to find a steady point 
which does not move when everything else is moving during an 
earthquake. For up and down motion, observe that in the last case 
just mentioned, W is like a steady point. 

When the forced and natural frequencies are nearly equal, we 
have the state of things which gives rise to resonance in acoustic 
instruments, which causes us to fear for suspension bridges or 
rolling ships. It would be easy to give twenty examples of im- 
portant ways in which this principle enters into practical scientific 
problems. 

124. Electrical Vibrations. Suppose a condenser of capacity 
k farads has the potential difference v volts between its coatings 
which are connected by a circuit of r ohms, of inductance I henries 
The quantity Q of electricity in the condenser is kv; the current 

mit of the condenser is c= --77= -k-jj, but this current is such 

at dv 

fir 

that «; = re + / ^. (See Art. 117.) 

(Xv 

TT 7^^ 77^^^ 77^^^ 7<^^ /^ /IX 

Hence "^^ -'^'k^.-ik-jr^ or ik-j^-\-rk^j-\-v = \) (1) 

dt df df dt ^ ' 

If we assume that in the circuit we have an alternator acting, or 
any other source of varying electromotive force E charging the 
condenser, we may write v- E instead of v in the above, and we find 

dH r dv V _E 
^^ d^'^ldi'^lk^lJc' 

r 1 

If we use 2/ for - and n^ for j^, we have 

dh ^rdv „ „„ .^. 

^ + 2/^ + ,^% = „^£^ (2) 

the very same formula that we had for mechanical vibrations. 



MAINLY ABOUT NATURAL VIBRATIONS 231 

It will be found that the analogy between mechanical and 

electrical vibrations is very close. Notice that 

W 
the mass or inertia — corresponds with inductance I. 

g 

The friction per foot per second h corresponds with the resist- 
ance r. The displacement x corresponds with voltage v or, to be 
seemingly more accurate, v i^ Q the electric displacement or, as 
some call it, the charge, divided by k. 

The yieldingness of the spring h corresponds with the capacity of 
the condenser k. 

The forced displacement y corresponds with the forced e.m.f. of 
the alternator.* 

As before, if we consider the natural vibration of the system left 
to itself with ^ = 0, if /has a value and it is less than n, if sln^ -P 
is called ^, we find that 

v = ^--«^sin(i?/! + e), (3) 

where A and e may have any values. 

Exercise. A condenser of ^' = 10"^ farads (called one one-hundredth 
of a microfarad) is short circuited through a resistance of r = 1 ohm f 
with an inductance Z=10"^ henries. What is the nature of the 
surging % 

Here 2/=y = 104, %2^-— J— -— = 10^2, so that 7i=106. If r 

is 0, the periodic time r=27r-j- 10^ = 6-28 x 10"^ second or the fre- 
quency is 159100 per second [the wave length of the radiation 
into space is the velocity of light 3 x 10^^ cm. per second divided 

by 159100 or it is 190000 cm. or M8 mi les]. 

As r is not 0, in fact /= 0-5 x 104,^ = n/7i2-/2 = 7ioi2 _ 0-25 x 10^, 
we may take it that 'p is really the same as n, so that the frequency 
is still 159100 per second. The answer is then, that v follows the 
law (3) or, what is a more suitable form, 

^_^^g-5000«cosl0% (4) 

* When E is 0, that is, when there are no forced vibrations, it is easy to 
see that instead of (1) we may write 

or lk-j-^ + rk-j- + c = 0. 

dt^ dt 

t In wireless telegraphy we have to recollect that there is loss of energy by 
radiation as well as ohmic loss, but in this elementary exercise we may assume 
that the ohmic loss includes the radiation loss. 



232 ELEMENTARY PRACTICAL MATHEMATICS 

where Vq is the voltage between the terminals of the condenser at 
the time which we choose to call 0. Or again, 

c = Coe-5«^'sinlO% (5)* 

a suitable form if we consider the surging current. 

If there are two tanks of water A and B connected by a large 
pipe; the water in A higher than the water in B by the height 
V feet; as soon as the communication is made, v follows a law 
like (4) and the current of water in the pipe follows a law like (5). 
Too much water comes into B and then it runs back again, the 
surges getting less and less as time goes on. 

It is evident that the periodic time of vibration of an electric 
circuit depends so little upon r that we may neglect r and say that 

71 = ^, and as n^ = jr, then T=lTrsJlk or the frequency = 1 -r ^tt^/A;, 

and wave length in the ether is 3 x 10^^ x ^irsjlk or 18-85 x lO^Vl 
centimetres or 1*17 x XO^slTk miles. 

125. A mechanical vibrating system P, if its frequency or pitch 
is anything from say 30 to 5000, acts on the air and sends out 
waves of sound; if these act upon another mechanical vibrating 
system Q they set up forced vibrations in Q. If Qs natural frequency 
is exactly the same as that of P, the forced vibration of Q may be 
very great. Sometimes when no musical sound is audible among 
the street noises a M'ire of a piano in one's room gives out an audible 
note because its frequency of oscillation happens to agree exactly 
with that of some inaudible musical note that has reached it. In 
such a case, it takes time for the forcing influence to produce the 
resonance ; indeed in our mathematical work we assume that the 
forcing influence has lasted for an infinite time. 

In the same way an electrical vibrating system P in Ireland, with 
a natural frequency of say 159100 per second (in the above example) 
sends out waves through the ether ; if these act upon another 
electrical system Q in America, they set up forced vibrations in Q. 
If Q's natural frequency is 159100 per second, Q gets a forced 
surging of current which may be great enough to affect instruments, 
and thus it is that we have wireless telegraphy. The condenser of 
the P system is regularly charged and discharged by sparks through 

*(4) and (5) differ very slightly as to the zero of time. As =c-k-^, if 

we assume (5) to be true we can easily find (4), so that there shall be exact 
correspondence, but is this worth while ? 



MAINLY ABOUT NATURAL VIBRATIONS 233 

the circuit already described, and each spark causes waves to be sent 
out whose amplitudes rapidly diminish ; in fact each set dies away 
in about one ten-thousandth of a second in the above case ; but at 
regular intervals we have fresh charging and discharging. It is 
extraordinary that in spite of these dyings away and renewals there 
is the sympathetic action of which I have spoken. Many inventors 
are now trying to find a more continuous transmission than what 
is now set up by sparks. It needs a sudden shock to start such 
vibrations, just as it needs the shock or blow of air against the lip 
of an organ pipe to set it going. The shock initiates vibrations of 
all periods, and with one of these the system synchronises, and thus 
it responds and magnifies. This is exactly analogous with the 
sounding of an organ pipe. 



CHAPTER XXX. 
FORCED VIBRATIONS. 

126. In what follows I shall assume that the natural vibrations 
of a system have been damped out, and they will not be considered. 

Let Arts. 121-123 be carefully read again about the body sus- 
pended by a spring, the upper support of which is displaced 
downwards through the distance y at the time t. Let y = asmqt. 
Equation (2) is 

-£ + 2f-£ + n^x = n^a8mqt (1) 

It is known that the forced motion is of the form x = A sin (qt + e), 
and therefore -^ is qix, -^ is qH^x or - q\ and hence (1) is 

(-q^ + '2fqi + n^)x = n^a sin qt, 

n-a sin qt 
n^ - (f- + 'Ifqi 

We know from Art. 119 that this means 

x^-A sin {qt - e), 

where A = , and tan e = ^^^ 



If we take //7i = 0*0707 as in Art. 122, and if we let qjn be called 
^, and let a=l, 

^ = 1-^7(1-^2)2 4.^2/50 and tane = 0-1414p/(l -i?2). 

Let the forced amplitude A be plotted with p. Let e also be 
plotted with j?. These results ought to be compared with those 
tabulated in Art. 123, where there was no frictional damping. In 
the frictionless case, e was either or 180°. If the study of these 



FORCED VIBRATIONS 



235 



p 


A 


e 


0-1 


roi 


0°-8 


0-5 


1-333 


5° -4 


0-8 


2-655 


17° -44 


0-9 


4 372 


33° -82 


0-95 


6-024 


54° -02 


0-97 


6-697 


66° -7 


0-98 


6-934 


74° -06 


0-99 


7 070 


81° -97 


1-00 


7-071 


90° 


101 


6-934 


98° 


103 


6-331 


112° -73 


110 


3-827 


143° -5 


1-5 


0-7879 


170° -37 


20 


0-3319 


174° -6 


5-0 


0-04167 


178° -32 


100 


0-0101 


179°-2 



results takes some weeks of time, it is probable that the time is well 
spent. The student ought to work another case, where there is more 
friction. 

127. In the three examples at the end of Chapter XXVIIL, I 
studied the electric currents which were forced on the system by an 
electromotive force or impressed voltage. In Chapter XXIX. we 
studied the currents which flow in electric circuits when they 
are left to themselves, the natural surgings which are rapidly 
damped out. I have returned now to the forced vibrations of 
systems. 

If V is the voltage between the coatings of an electric condenser 
of capacity k, the charge in the condenser isQ = kv, and the current c 

into the condenser is G = k—. If «? is of the form a sin (qt -he), we 

7 Ctt 

know that -in is qiv and c = kqiv. Now if B is the resistance of a 

contrivance, the current C flowing because of the voltage Fis C=-t,- 

We see in the case of the condenser that g = v-^ t— ., so we may say 

that a condenser has a resistance ^j— .. 

kqi 

Multiplying numerator and denominator by i, and remembering 



that i^=\, we find the resistance of a condenser to be 



kq 



I have 



already used this idea in Arts. 118 and 119. 

We know that a circuit which is of resistance B ohms, inductance 



236 ELEMENTARY PRACTICAL MATHEMATICS 

L henries, and which has a condenser of K farads in series, may be 
said to have the total resistance 



Notice that when the i term is 0, we may expect large cuiTcnts. 



B+Lqi-jir- or E + i{ 



Ex. 1. In fact, we say that such a circuit is in tune when 
Lq = ^ or LKq^= 1. Suppose we wish it to be in tune for a fre- 
quency /, so that q = 27r/= 6000 say [this would be a frequency of 
about 955]. Then L/iT x 36 x 10^ = 1. 

Let us take K= 10"^ or one microfarad and ^ = ^7^. 

ob 

Fig. 43 shows a part of a circuit with a potential difference v 



->^55silJLOJLlQJULOiLQJL 



L 

Fig. 43. 



between its ends. Let us take it = 100 ohms, which is rather high 
for such a circuit. Let the voltage be 

■v=li\4,smqt. 

Then the current hv-^ resistance or 

1414 sin g^ 



100 + 



V36 q ) 



For the following values of q^ I give the current: 



<1 


Resistance. 


c 


3000 


100-250i 


5-252 sin (g« + 69°) 


4500 


100 - 97» 


10-152 sin (g< + 44°) 


6000 


100 


14-14 sing^ 


7500 


100 + 7o^ 


11 -312 sin ((7^- 37°) 


9000 


100+139i 


8"255sin(g<-54°-5) 


10500 


100+196i 


6-427 sin (g«- 63°) 



If I had taken a smaller R, the greatness of the current when 
q = 6000 would be more marked. The student ought to plot such 
answers on squared paper, showing how both the amplitude of c and 
also its lag or lead alters as q alters. 

If we had no ohmic resistance the current would be infinite for 
2 = 6000. 



FORCED VIBRATIONS 



237 



Ex. 2. Two circuits in parallel. What is the total current? 
One has a current C2, ohmic resistance R ohms, and inductance L. 
The other has a current Cg and is merely a condenser of capacity K 
(see Fig. 44). 



Co = 



^~R^-LqV 



r^ . ^ 1-KLq^ + EKqi 
c^ = Kqiv, C = c^ + Cq = ^ ^ . — —V. 



R + Lqi 



* It is evident that C will be small if KLq^ =1 or ^ = -j^, and if 
we depart from this value of q, we find C to be large. v AL 

R L 




Fig. 44. 



I will take a case of very great inductance, and I will tune the 
arrangement so that KLf=\. Let R = 0% Z = 0-01, 7^=4 x 10-<^, 
so that the critical q is 5000 [a frequency of about 800]. I will 
only give the amplitude of C. Let v = 1000 sin qt. 

Notice the smallness of C for ^ = 5000. 



1 


Amplitude of C. 


3000 


21-30 


4000 


9-00 


4500 


4-22 


5000 


0-20 


5500 


3-82 


6000 


7-33 


8000 


19-50 


10000 


30-00 



If I had calculated c^ and Cg both of them would have been large 
when q = 5000 ; their sum C is small because they differ in phase 
nearly 180°. In the same way the sum (or resultant, as it is called) 
of two great forces may be small when the forces are nearly opposed 
to one another. 

Ex. 3. If a student will draw curves showing how c of Ex. 1 
and C of Ex. 2 vary as q is altered, these curves will suggest to 
him Mr. Sidney Brown's old invention to cause a complex current 
which is the sum of many simple sine functions to divide itself, so 
that that part which is of nearly one frequency shall go along 
one circuit and all the rest of the complex current shall go in the 
other. 



238 ELEMENTARY PRACTICAL MATHEMATICS 



Suppose we give the total current (Fig. 45) three ways or circuits 
of passing from a point P to a point Q, there being the voltage v 



established between P and Q. The 




first circuit has r, I, and a 
capacity k, so that its total 

resistance is ri-illq-j-j 

and the current in it 
V 



is 



r-\-i 



iH) 



Fig. 45. 



The second circuit 
B and L, so that its resistance is B + Lqi and its current is 

F 
^^ E + Lqi' 
The third circuit has a condenser K and its current is 

Cg = KqiF. 
Let Cg + Cg be called C, then 

1 



has 



-( 



E + Lqi 



+ Kqi)F. 



{1-KLq^-h 



IiKqi)\r + i(lq 



Hence ^i — j— ^ 

c E + Lqi 

Now Brown desires that C shall be exceedingly small and c great 
for a particular value of q, say q = 5000, but for all other values of q 
he desires that C shall be great and c small. 

It is evident that for q = 5000 we ought to make KLq- = 1 = Mq^, 
as this makes the numerator small. Hence KL = kl = 4: x 10"^. As 
Brown wished to have a telephone of considerable resistance in 
the c circuit, he was compelled to use a large I and a large r ; take 
then r=100, 1 = 4=; therefore A; = 10"^. But he could have the B 
and L of his other circuit small, so he took ^ = 0*5, Z = 0-01; 
therefore ^=4x 10"^. Calculate the following results for various 
values of q : 



1 


c 

c 


3000 


455 


4000 


81-02 


4900 


0-6778 


5000 


0200 


5010 


0-0278 


6000 


53-74 


7000 


188-1 


10000 


900-1 



FORCED VIBRATIONS 239 

We need not trouble about anything but amplitudes. We see 
that if ^ = 5000, c is 50 times as great as C; if ^ = 4000 or 6000, c is 
from -^^ to -gjj of C ; thus Brown's object is effected. 

If we assume that R and r are so small as to be negligible, we get 

C 
a much simpler expression for - to study and far more striking 

figures. 

Letting the circuit of Ex. 1, Fig. 43, be called I., let the compound 
circuit of Ex. 2, Fig. 44, be called 11. Mr. Brown simply shunted 
a circuit I. by a circuit II. 

128. The electrical exercises already given are all examples of 

the following general rule. Let 6 stand for -^. A current having 

the ohmic resistance r, the inductance /, and the capacity k may be 

said to have the resistance ^ + ^^ + t:^- In any network of conductors 

conveying electric currents, if there is a constant electromotive 
force E in any branch or if a potential difference of voltage V be 
established between any two points, then the constant current in 
any branch can be calculated, being E ov V multiplied by an 
algebraic expression involving all the resistances of all the branches 
rj, rg, ^3, etc. If E ov V \^ varying, we shall use the same algebraic 
expression ; but now, instead of a mere ohmic resistance ?3, for example, 

and ^'3 a capacity in that branch. However complex the expression 

may be, when cleared of fractions, etc. (we treat 9 as if it were a 

mere algebraic quantity ■^), it simplifies to this, that an operation like 

a + he + cd^-\- dO^ -f e(94 +/6>5 + etc. ... 

a' + b'd + c'd^ + d'e^ + e'e^+fe^ + etc. ^ ^ 

has to be performed upon some voltage which is a function of the 
time. If the function is of the type sin qt, we substitute qi for 
in (1), and the complicated operation becomes 
a 4- bqi + cqH^ + dqH^ + eqH'^ +fqH^ + etc. 
a' + h'qi + c'qH'^ + d'qH^ + e'qH"^ +fqH^ + etc. 

_ a - cq^ + eq'^ + etc. +iq{b- d(f -^fq^ - etc. ) 
~ a! - c'q^ + ey + etc. + iq{b' - d'q^ +fq'^ - etc.) * 

We may call this j— , and we know from Art. 119 the effect 

of such an operator. 

[* The proof of this is easy but tedious ; it is given in my CcUcidtts, pages 231-4.] 



we use ^3 + ls^ + Trh if h is the ohmic resistance, l^ the inductance. 



240 ELEMENTARY PRACTICAL MATHEMATICS 

If two circuits r^, r^ are in parallel with constant currents Cj and c^, 
the whole current C^c^ + c^ divides itself so that 

^ ^2_ and ^2_ ^1 



G r^ + r^ rj + rg 

Also the combined resistance of two circuits in parallel is 



i\ + r^ 
so that if V is the voltage 






V. 



If the current is alternating, of the type sin^^, we use these same 
expressions, only that i\ and r^ are now unreal quantities. Thus, 
suppose we use no condensers ; take r^ + \qi instead of i\ and 
r^-\-l^qi instead of rg, and see what each formula becomes. Now 
take such an example as rj = 1, /j = 0'l ; ?-2= 100, ?2 = ^*^^ j calculate 
the value of each of the above formulae for many values of q, and 
you will become acquainted with very interesting results. 

Example. Or we may proceed as in Ex. 2 above. I will, 

however, take different numbers. Let ?;= 1414 sin g-^. Let one 

circuit have a resistance 100 + ^z (that is i^= 100, L=\). Then for 
? = 1000, 

'2 = lMTlS ^' ^2= 1-407 sin (1000^ -84°-283). 

Let the other circuit have a condenser, merely, one microfarad in 
capacity or K^ 10~^. Then Cg = Kqiv ; that is, 

C3= 1-414 sin (1000^ + 90°), 
and the total current C =03 + 63 is 

a = 0-1407 sin (1000^ + 5°'7l7). 

It may disturb the mind of a beginner to find a great current in 
each branch but the total current small ; but, as I said before, it 
is analogous with the sum of two great, nearly equal, but nearly 
opposite forces. In such a case as the above 1 usually say that the 
compound circuit is in tune with q= 1000 [a frequency of about 160]. 
All the exercises given in my Calculus, all the exercises that can be 
set on alternating currents through given coils and condensers, 
merely depend upon these simple formulae which are so well known 
to us for constant currents. 

129. I have only to add here the effect of mutual induction 
between two circuits. If there are two parts of circuits r^ and r^ 

with mutual induction m between them, let i\ stand for '*i + ^1^ + r-^ 



FORCED VIBRATIONS 241 

and r., for r^ + 1.^& + r-z > ov \i either circuit is without a condenser, 

let its Tn term be 0. Let v^ be the voltage between the ends of 

the first circuit and v^ between the ends of the second circuit. Then 

v^ = r^G^-Vmec^ n^^ 

So if we are given v^ and i?.,? we can calculate Cj and Cg, and we can 
work a great many other more troublesome-looking and more 
curious problems. 

m may be either negative or positive. 

If the currents are of the type sin^^, of course we substitute qi 
for d in our work. 

For examples of ordinary transformers I must refer the student 
to my Calculus. He will have no difficulty in working any of the 

exercises there given. He will note that I there use for -j-. 

Perhaps he had better do as I have done in this book, use qi every- 
where instead of or — • but the actual numerical work is just 
the same. 

130. It will be noticed in the above exercises that when I have a 
circuit containing r and / or r and / and k, I speak of the resistances 

as being r + Iqi and r + illq-j-\ Some people call these impedances. 

It is of no consequence which name is given, but strictly speaking 
the impedance is the amplitude part of r -f- Iqi or Jf^ + l^q^. 

For the transformers used in wireless telegraphy, and sometimes 
in telephones, we do not have the simple rules so familiar to the 
electrical engineer, because condensers may be in the circuits and 
also 1-J2 - wi^ is very far from being 0. It may be well to give the 
following example from wireless telegraphy. 

131. Systems influencing each other. Let two pendulums, A 
and B, of very nearly the same length, be hung from a thin 
horizontal rod or string ; or let them be hung from rigid supports, 
but sufficiently near .one another to exercise force on one another, 
either by means of india-rubber threads, or even, without such 
threads, let them affect each other by the mere currents of air that 
they produce. If A is vibrating and B at rest, B begins to 

P.M. Q 



242 ELEMENTARY PRACTICAL MATHEMATICS 

vibrate, and its swing increases to a maximum and then diminishes, 

whereas the swing of A decreases nearly to and then increases 

again, and this exchange of energy continually goes on between 

them. In any of these pendulum cases it is easy to state the 

equations of motion and solve them. But I prefer to give the 

analogous case of two electrical circuits in presence of each other, 

each with its resistance, inductance, and capacity, left to themselves ; 

that is, there is no applied electromotive force in either. 

-1 J 

If r stands for r + Z^ + j-?,, where 9 is -j-., if m is the mutual 
Ku at 

induction, the two circuits and the currents in them being dis- 
tinguished by the suffixes 1 and 2 ; then (1) of Art. 129 is 
r^Cj + mBc^ = = r^c^ + rndc-^ , 

T C 

and we have {iYj^ - 171^9^)6^ = 0, Cg = - -^. 

It is easy to develop these equations. The expressions are 
simplified by taking the ohmic resistances to be 0, as they usually 
are insignificant in comparison with the inductance terms. We 
shall therefore have no damping in our oscillations. The result is 
of the form (S'^-^- A9^ + B)c^ = 0, 

and we have exactly the same equation for c^. The auxiliary 
equation (2) of Art. 133 being x^ + Ax^ + B = 0, we find roots of the 
form x= ±ai and x= ± /Si, so that the answer for Cj is 

q = M^ sin (at + e^) + iVj sin (jSt + g^), 
where M^ and iV^ are arbitrary constants and 

Cg = ilf 2 sin (o-i + ^2) + -^2 si^ (/^^ + ^2)- 
In fact, Cj has two frequencies and c^ has the same two frequencies. 
It will be found that 



4' 



2k^k^{\l^-m^) 

gives the value a if the plus sign is taken and the value P if the 

minus sign is taken. 

We may write this differently. Let t-^ = 2TrsJk-^\ be the natural 

period of the first circuit and let t<^ = ^irjkji,^ be the natural period 

of the second circuit when they are quite separate from one another. 

, 27r 27r 

Let fi^ = iir'^m s/k^k^. Let T' = — and T"=-q be the two periodic 



FORCED VIBRATIONS 243 

times Avhich we find in c^ and C2. Then 



i. 



'2t^V - 2/x^ 



is the value of T' if the plus sign is taken, and it is the value of T" 
if the minus sign be taken. All the above work is easy algebra, 
which any student can do for himself. 

The usual case is to let l-Jc-^ = Ijc.j, so that t^ = t^ = T say. We 
find now that ^,^^^^--,^^ T ==sIT^~^K 

If, now, there is loom coupling as it is called, that is, m^ much less 
than Zj/2 (oi* ^^® have much magnetic leakage), fi^ will be small 
compared with T, so that T' and T" are nearly equal. 

Example. Let l^ = lO'^ henry, /, = 20 x IQ-^, w = IQ-^, Jc^ = lO'^ 
farad, k., = 0-05 x IQ-^ farad. 

r=27rx/y^ = 27rv/y;= 19-90x10-8, /x2 = 8-83x 10-16; 

r = 20-125xlO-8 and r'= 19-67 x lO-s, 

and a = 31-23xl06, ^ = 31-94x106. 

We have then exactly the effect of beats in music when two notes 

of different pitches (we call them frequencies) are sounding together. 

Each term in c^ is connected with the corresponding term in Cj by 

the relation - c„ = -^ or ^77-.. In this case we find that 

if2 = 0-l736ifi and N^ = ()'^\nN^. 

The algebra gets troublesome if we proceed further. But the 
general result is obvious enough if we look at the two swinging 
pendulums which I have described. If the pendulums are of 
exactly the same length, the illustration is very perfect. If one of 
the pendulums, when its swings are small, is quite stopped, the 
vibrations of the other no longer increase and diminish. This 
gives the reason why the quenched spark method is important in 
wireless telegraphy. 



CHAPTER XXXI. 

PERIODIC FUNCTIONS IN GENERAL. 

132. I have considered the simplest kind of periodic function in 
Chap. XXVIII. A periodic function of the time is one which 
becomes the same in every particular (its actual value, its rate of 
increase, etc.) after a time T. Thus T is called the periodic time, 
and its reciprocal is the frequency. Algebraically the definition of 
a periodic function is xf^^ ^ xr^ ^^ji^ 

where n is any positive or negative integer. 

Fourier's Theorem can be proved to be true. It states that any 
periodic function whose complete period is T being called f(t), and 
q being 27r/T, may be expressed as 

X =/(0 = Aq + J^sin (qt + e^) + Ac^sin {2qt + e^) 

+ ^3sin (85^^ + 63) + etc (1) 

In the same way, the note of any organ pipe or fiddle string or 
other musical instrument consists of a fundamental tone and its 
overtones. We know from Art. 116 that (1) is really the same as 

f(t) = ^0 + ^1^^^ 2^ + h^ao^ qt + a^Bin 2qt + h^cos 2qt 

+ ttgsin 3qt + b^cos Sqt + etc., (2) 

if A-^^ = a-^^ + b-^^ and tan 61 = ^, etc. , v 

In all our forced vibratiottitproblems in mechanics or electricity, 
we have to operate with functions of -^ or 6 upon a function of the 

time/(0. In every case I assumed this to be of the shape asing^. 
But if it is any periodic function whatsoever, and if we express it in 
the form (1), we operate upon each term of (1) as if it alone existed, 
and our complete answer is the sum of these partial answers. 



PERIODIC FUNCTIONS IN GENERAL 245 

For the theory of the Fourier development I must uefer students 
to my Calculus. 

In valve motion work and in electrical work, it is very important 
when given a curve showing f(t) and t to be able to develop any 
periodic function in the forms (1) or (2). I described a graphical 
method in the Electrician newspaper of June 28*^, 1895. In the 
same paper on Feb. o*^, 1892, I described a method which might be 
used when a table of numbers is given for the equidistant values of 
f{t). Prof. Henrici's analysers are machines which give the correct 
results when we turn a handle. The following method is probably 
the quickest when we have tabulated values. We have greater and 
greater accuracy when we have more and more tabulated values. 

Example. I here tabulate 24 equidistant values of x, and I 
assume that A-^^, e-^, A^, e.^, A^, e^, A^, and e^ have to be found. 
I had better use the letter <^ for the angle qt ; evidently 4> passes 
from to 360° or from to 2^ when t passes from to T, that is 
through the whole period. First, add the 24 ordinates together, and 
divide by 24 ; we thus get A^=-b. We now subtract 5 from every 
ordinate and call the result x. In this particular case I know that 
there are not more than four components. Then 

x' = ^jsin (</> + ^i) 4- A^^m ('2cf> + e^) + A^sin (3<^ + e^) 
+ A^sm{4.(l> + e^). 

Let the student imagine x' plotted vertically and (f> borizontally 
on squared paper from ^ = to </> = 360°. Then if one half of the 
curve from 180° to 360° is superimposed on the other half from 0° 
to 180°, the P* and 3"^ components will be eliminated if correspond- 
ing ordinates are added [this is easy to see if anyone of these 
components be drawn separately with any value of e], and the 
resulting curve will be 

x" = 2[^2sin (2<^ + e^) + A^sin (4<^ + e j]. 

Similarly if the original curve be divided into three equal portions 
by lines perpendicular to the axis of </>, and the three parts be 
superimposed and corresponding ordinates added, the 2"'' and 4"' 
components will be eliminated, and the resulting curve will be 

x'" = Sffgsin (3<^ + gg). 

It is an easy exercise for the student to prove this either graphi- 
cally or analytically. If he has difficulty let him consult Mr. 
Wedmore's paper in the Proceedings of the Institution of Electrical 
Engineers, 1896. The table shows how the above method is used 
without drawing the curves. For instance, columns A, I, and / are 
the three equal parts superimposed, and when added give column K, 
which is 3 times component 3 ; in this case zero. 



246 ELEMENTARY PRACTICAL MATHEMATICS 

An examination of the table easily shows how it is all produced. 

Com2)oiient 1. Imagine column N to be continued to the top of 
the table; ordinate will be 2 -5 20; average of ordinates from to 





1 
1 


X 


A 


B 


G 


D 


E 


F 


G 


H 


*° 


x' 

or 

x-h. 


This is 
A 

super- 
imposed 
on itself. 


A+B. 


Half of 
C being 
sum of 
com- 
ponents 
2 and 4. 


Being D 
super- 
imposed 
on itself. 


Being 
D+E. 


or com- 
ponent 4. 


D-G 
being 
com- 
ponent 2. 








8-02 


3 02 


-202 


100 


0-50 


-0-50 


0-00 


0-00 


0-50 


15 


1 


8-37 


3-37 


-2-33 


104 


0-52 


-0-345 


0-175 


0875 


0-4325 


30 


2 


8-33 


3-33 


-2-66 


0-67 


0-335 


-0 165 


0-170 


085 


0-250 


45 


3 


7-93 


2-93 


-2-93 


0-00 


0-00 


0-00 


000 





00 


60 


4 


7-34 


2-34 


-301 


-0-67 


-0-335 


0-165 


-0170 


-0-085 


-0-250 


75 


5 
6 


6-71 
613 


1-71 
113 


-2-75 
-213 


-1-04 
-100 


-0-52 
-0-50 


0-345 


-0-175 


-0-0875 


- 0-4325 


• 90 






0-00 


-0-50 


105 


7 


5-58 


0-58 


-1-27 


-0-69 


-0-345 






0-0875 


0-4325 


120 


8 


4-99 


-001 


-0-32 


-0-33 


-0165 




0-085 


-0-250 


135 


9 


4-38 


-0-62 


0-62 


0-00 


0-00 






0-00 


00 


150 


10 


3-80 


-1-20 


1-53 


0-33 


0-165 


Kr- 


-0-085 


0-250 


165 


11 
12 


3-34 

2-98 


-1-66 
-2 02 


2-35 


- 0-69 


0-345 






-0-0875 


0-4325 


180 








0-50 


-2-5-20 




195 


15 


2-67 


-2-33 


M 


' continued iipwa 


[■ds 


0-52 


- 2-850 




210 


14 


2-34 


-2-66 




^ 




0-335 


-2-995 




225 


15 
16 


2-07 
1-99 


-2-93 
-3-01 






0-00 
-0-335 


-2-930 
-2-675 




240 


-001 


302 









255 


17 


2-25 


-2-75 


-0-62 


3-37 







-0-52 


-2-230 




270 


18 


2-87 


-213 


-1-20 


3-33 







-0-50 


-1-630 




285 


19 


3-73 


-1-27 


-1-66 


2-93 







-0-345 


-0-9-25 




300 


20 


4-68 


-0-32 


-202 


2-34 







-0-165 


-0-155 




315 


21 


5-62 


0-62 


-2-33 


1-71 







0-00 


0-620 




330 


22 


6-53 


1-53 


-2-66 


113 







0-165 


1-365 




345 


23 


7-35 


2-35 


-2-93 


0-58 







0-345 


2-005 




Mean 
Ordinate 


}- 




Being 

super- 
imposed. 


Being 

A 
super- 
imposed. 


A+I+J 
being 
3 times 
com- 
ponent 3, 
which 
isO. 




D 
repeated. 


A-D 
being 
com- 
ponents 
land 3 
or comp. 
1 only. 








A 


/ 


./ 


K 




M 


N 





No. 1 1 inclusive, treating all as positive, is 22*900 -^ 1 2 = 1 -908. We 
use this method of finding A-^ because of the rule (see Ex. 2, 
Art. 56) : 



Maximum ordinate A-^ = 1 -908 x ^ = 2-997, say 3. 



PERIODIC FUNCTIONS IN GENERAL 247 

rr . sin e. 2-520 n o i ^i. r k^to 

^"^ ^^* 'i' sirTgo" == 2^997 ^ ^'^^ ^ therefore e^ = 57 . 

Component 2. By inspection of column ZT, the maximum ordinate 
is 0-50, and 62 = ^^0°. 

Component 3. Zero. See column K. 

Component 4. Average of ordinates from to No. 2 inclusive is 

0-1725 -^3 or 0-0575, and ^4 = 0-0575x^ = 0-091. 

By inspection e^ = 0. 

Hence a; = 5 + 3 sin (</> + 57°) + 0-5 cos 2<^ + 0-09 sin i<f>. 

Note. When we have few ordinates of a sine curve it is not exact enough 
to take their mean value (taking them all as positive as I do above) as the mean 
ordinate of the real curve. I am not satisfied in such a case unless I plot the 
ordinates and draw the probable curve to give me A and e. 

133. A differential equation like 

d'^x j^d^x ^d^x r,dx ^ rr /i\ 

iF+^^+«w+-^*+^^=^ (1) 

if P, Q, R, S, and T are functions of t only, or constants, is said 
to be a linear differential equation. 

Suppose we put T= ; we shall find a solution x = some function 
of f. Let us write it x=f(t). It will involve four arbitrary con- 
stants, because the given equation is of the fourth order. Now, if 
when T is not we giiess at one solution and find it to be a; = some 
other function of t, which I shall call F{t), then it is easily proved 
that the most complete solution of (1) is 

x=f(t) + F(t), 
I shall here consider only cases in which P, Q, B, and S are 
constants. First, let T be 0. Assume that y = ilfe'"' is a solution, 
and we see that it is so, no matter what M may be, if 

m^ + Pm^ + Qm^ + Em + S=0 (2) 

This is called the auxiliary equation. An equation of the fourth 
degree has four roots. Suppose these to be the values of m which 
satisfy the equation (2), a, b, a + p% a- pi; the answer x =f(t) is 

X = Ae""' + Be^' + Ce<'^+^')< + De^''-^'^\ 

where A, B, C, and I) are arbitrary constants. 

As C and D may be unreal, and as e'^'* may be written 
cos pt + i sin pt 



248 ELEMENTARY PRACTICAL MATHEMATICS 

(see Art. 137), it is easy to see that the solution most suitable for 
actual problems is 

X = Ae''* + Be^^ + e*' (71/ cos pt + iYsin (it\ 
where A^ B, M and N are arbitrary constants, all real. 

In Art. 121 1 use this rule in the solution of an equation of the 
second order. A real root of (2) of the form m = a gives rise to a 
term Ae^^ in the answer ; unreal roots of the form a ± ^i give rise 
to the terms e** (if cos pt + N^in pt). 

There are rules about equal roots, but they are seldom needed. 
The theory will be found in my Calculus. If, for example, x is 
displacement of a body and t is time, the solution x =f(t) is called 
the free or natural motion of the system, and the part x = F{t) is 
called the forced motion. As I have said at Art. 120, we often 
neglect the free motion as it has damping terms, causing it to quickly 
cease to exist. 

Excellent easy exercises on this subject will be found in («) my 
" Theory of the Himting of a Steam Engine," in the Appendix to my 
book on Steam, (b) the critical speeds of shafts with or without 
heavy wheels and the critical speeds of crank shafts, given in the 
Appendix to my book on Applied Mechanics. 

If in Art. 121 we imagine a series of several springs and weights, 
we get a system of several natural frequencies ; how will it behave 
when subjected to varying forces acting on the weights (Applied 
Mechanics, Appendix)^ It is exceedingly interesting to watch the 
natural vibrations of such a system or its equivalent torsional 
system, and it is easy to study mathematically. Lord Kelvin was 
fond of this study, and made much use of his results in his famous 
Baltimore Lectures on "Light." The work is so easy that I am 
tempted to give it here, but this book is getting too large. 



CHAPTER XXXII. 
EXTENDED RULES AND PROOFS. 

134. A knowledge of the differ en tiation of the few functions 
given in Art. 94 will suffice for nearly all the mathematical work 
that has to be done by the engineer. I mean by engineer, any man 
who applies the principles of Natural Science. 

By means of a few rules it is easy to become able to differentiate 
any algebraic function of x and to integrate a great many. 

Students often learn no more of this wonderful subject than to 
acquire these rules, and they rapidly lose their power after they have 
passed certain examinations, because they never have learnt really 
what dyldx means and they have taken no interest in the subject. 

I hope that our teaching will give students that kind of interest 
in the subject which is generally wanting, and that some of these 
students will pursue the subject in more orthodox treatises. I think, 
however, that they had better first study my book, The Calculus. 

1 . Let ?/ = ii + V + ^y be the sum of any three functions of x. 

Let X become x-v^x, and in consequence let u become \i-^hi^ 
V become v + 8v, and w become w-^t-hio. It results that if y becomes 

y^^y^ 8y = 8u + &v + &w 

8y _8u Sv 8w 

8x 8x 8x 8x^ 

1 . ^, ,. .^ dy du dv dw 

and ni the limit -/■ = -7- + -r + -i-- 

dx dx dx dx 

In Art. 93 I assumed this without proof; that the differential 
coefficient of a sum of functions is equal to the sum of the differential 
coefficients of each. 

A great extension of our table, Art. 94, may be made if the 
following rules are known. 

2. Differential coefficient of a product of two functions. 



250 ELEMENTARY PRACTICAL MATHEMATICS 

Let y = uv, when u and v are functions of x. When x becomes 
x + Sx, let y^8y=::{u + 8u)(v-\-8v) = uv + u.Sv + v.8u + 8u.Sv. 
Subtracting, we find 

8y = u.8v-hv .8u + 8u.8v 
, 8y 8v 8u ^ 8v 

^"•^ ^="&+''&+^»&- 

We now imagine 8x, and in consequence (for this is always assumed 
in our work) 8u, 8v, and 8y to get smaller and smaller without limit. 

Consequently, whatever -y- may be, 8w ^ must in the limit become 



0, and hence 



dx 



dy _ dv du 
dx ~ dx dx' 



The student ought to manufacture such exercises as this. 
Let u = 5;i'^ and v = 2i«^, 

y- = 8.#, -J- = 15a;2, so that, by the rule, 

^^ = bx^ X ^x? + 2x^x1 5a;2 = 1W, 
dx 

But as ?/ = lOa;^, we know that ~- = 1W. 

3. Differential coefficient of a ciuotient. 

Let y = -, where u and v are functions of ic. 

i hen y + oy = r-. 

Subtract, and we find * 

u + 8u u v.8u-u.8v 



^ v + 8v V v^ + v.8v 
8u 8v 
8y 8x 8x 
8x~ v^ + v.8v 

Letting 8a; get smaller and smaller without limit, v . 8v becomes 0, 
and we have du dv 

dy dx dx 

dx v^ 

or in words, " Denominator into differential coefficient of numerator, 
minus numerator into differential coefficient of denominator, divided 
by denominator squared." 

A few illustrations are easily manufactured. Thus, let 
u = 24:x'^ and v = 3x^ or let u=\6x'^ and v = 3x^. 



EXTENDED RULES AND PROOFS 



251 



135. If y is given as a function of z and z is given as a function 
of X, of course y/ is a function of x. Under such circumstances, if 
instead of x we take x-\-^x, and so calculate z + ^z, and with this 
saine z + ^z we calculate y + 8y, then we can say that our Sy is in 
consequence of our 8x, and 

8y 8y 8z 

8x 8z 8x 



(1) 



On the supposition that the two things written as 8z remain the 
same however small they become, we see that the rule (1) is true 
even when 8x gets smaller and smaller without limit, or 

dj^dy^ dz ^2) 

dx dz dx 

This is an exceedingly important rule, and the student ought to 
illustrate it in ways of his own. I might suggest drawing three 
curves, a first showing ^ as a function of Zj a second showing ^ as a 
function of x, and a third showing y as a function of x. The student 
ought to see clearly that at corresponding points the slope of 
the y, X curve is equal to the product of the slopes of the other 
two. 

The following examples are applications of the rule (2) 

Ex. 1. Let y = e''''. Put it in the shape y==e'\ u = ax, 
dv „ du 



du 



= e 



dx 



= a ; 



therefore 



dy 
dx 



e" X a = ae""" (see Art. 1 1 3). 



Ex. 2. Let y = log (a 4- hx) or let y = log u, where u = a + bx, 
dy 1 du , 



so that 



du u dx 



dx u a + bx' 



Ex. 3. Let y = sin (ax + b) or let y = sin ii, where u = ax-\- b, 



dy 



du 



:77. = cosw, ^ = a, 



so that 'T^^ ^^^ u = a cos {ax + b). 

Similarly, \iy = cos {ax + b), -^ = - a sin (ax + b). 



252 ELEMENTARY PRACTICAL MATHEMATICS 

136. The following examples are not needed for any of the work 
of these classes, nevertheless I am weak enough to give them : 

Ex. 1. Let ^ = tana: = . By the rule for a quotient, we have 

cos X 

c??/_cosa:.cosic-sina;( -sina;)_ 1 

dx ~ cos^ic cos'-^a;' 

Also, if y = tarn ax, ^= -„ — . 

dx cos^ax 

Ex. 2. If X = a<fi-a sin ^ and y = a — a cos ^, where x and y are 
the co-ordinates of points on a cycloid, 

dy dy ^ dx a sin <^ _ sin </> 
dx~ d(f) ' d4>~ a-a cos <j) 1 - cos 4> 

Ex. 3. Let y = sin~ia;; in words, y is the angle whose sine is x. 
Hence a; = sin^, y- = cos ?/ = Vl - sin^y = >/l - a;'^. 

Hence ^ = -7-=^ • 

dx v/l-a;2 

Similarly, if y = sin~i -, then -f^ = . — . 
a ax isja^ _ x^ 

Prove that if y = t3in~^x, then ~l=~ -; 

ax 1 "T ic 

and that if ^ = tan~i-, then '^ 



a dx a- + x^ 

Ex. 4. If ^ = a"", then log y = x log a. 
Differentiating with regard to y, we have 

1 dx, 

- = -r loff a. 

y dy ^ 

-j-=^y log a = a* log a. 
Ex. 5. If y — xJxP'-a^^ or y = vJ^y where u = x^-a^^, 

du 2^ ' dx'"^' 

so that ^J = \-uHx = -=l=. 

dx 2 Va;2 - «« 

Ex. 6. Example of a product. Let ?/ = e'*'sin(ia; + c). 

^ I . .. dy dv du 

Our rule is, if y = wz;, j^ = '^ t- + ^^ j^- 



EXTENDED RULES AND PROOFS 253 

Here u = e'^, -j- = ae'^, 

dv 
V = sin (hx + c), -^ = h cos {l)X + c). 

Hence -~- = be'^cos {hx + c) + ae"*sin {hx + c) 

or -^ = e**{5cos(6a; + ^)4-asin(ia; + c)}. 

Ex. 7. Or let y = e''''(/4 sin hx + B cos &a;). 

-^ = e'^{Ah cos ^.7j - i?6 sin hx) + ae"''{A sin 6a; + i? cos J.'j;), 

or ^ = e*^ { («^ - ^*) sin ^>a; + (^^> + aB) cos 6a:} . 

Ex. 8. Show that y = e^sm bx or e^cos bx satisfies the equation 

Ex. 9. Show that ic = e~ ^(^ sin 5^ + ^ cos bt) satisfies the equation 
(^x ^dx _, _ 



EXERCISES. 

The average value of y from x=a to x=h is the area / y. c?.r divided by 
h-a. -^^ 

A knowledge of some very elementary trigonometrical formulae (those 
of Exercise 3, Art. 28) is necessary for the working of some of the examples 
in this page. 

1. The average value of acos(5'^ + e) or asin(g'^ + ^) for the whole 
periodic time T is zero. 

2. The average value of a^uiqtxhsm{qt±e) for the whole periodic 

, . . ah 
time IS -—cose. 

2 

3. The average value of acoBqtxhco^{qt±e) for the whole periodic 

,. . ah 
time IS — cose. 
2 

4. The average value of the square of a sin qt for the whole periodic 
time is a'^j% (See Art. 118, note.) 

5. The effective value of a current 6'= a sin ^^ is the square root of 

the average value of the square of C; that is, the effective current = -— 
'or 0-707a. v2 



254 ELEMENTARY PRACTICAL MATHEMATICS 

Thus, if C= 1414: sin qt, the electrical engineer says that the current is 
1000, because he speaks of the effective current. So also with voltage. 

It follows from (2) or (3) that the effective amperes x effective volts x 
cosine of the angle of lag or of lead between amperes and volts is electric 
power in watts, and cose is what is called the load factor. 

137. In the next two articles I propose to give Proofs which I 
discovered when this book was in the hands of the printer. 

Some students may be dissatisfied because in Art. 95 I assumed 
the Binomial and Exponential theorems without proof, and they 
may complain of want of rigour, generally, in my methods. But 
from the beginning I have had to show that every student must 
first make acquaintance with formulae by much exercise work ; he 
must then learn to use calculus methods ; afterwards even the man 
of non-mathematical mind will probably be interested in rigorous 
proofs. It must be understood that no elementary student, however 
mathematical, however clever, is introduced to proofs whose rigour 
will satisfy the great mathematicians. The following scheme is as 
rigorous as that which is usually given ; it is much simpler because 
I use the calculus idea from the beginning. First, as in Arts. 134-5, 
prove the simple rule for the differentiation of a product, and that 
if u is a function of x, ^y ^y ^^^ 

dx du ' dx' 

138. To show that when ti -^x^. -^ = nx^~'^. It can be shown at 

■^ dx 

once (as in Art. 89) that this is true when ?i is 2 or 3. I shall 
first prove that it must be true when n is any positive integer. 

(a) Suppose the rule true when 7i = r; I will show that it must 
also be true when n = r+l. 

Take y ■= a;''^^ = xxx''. By the rule for a product, we have 

-^ = a;'* + rx""'^ xx = (r+ !)«*". 

As the rule is true for r = 2, we see that it must be true for ?• ^ 3 
or 4 or any positive integer. 

- 

(b) If y = x^, where p and q are any positive integers, then 

y'^ = x^ = z say. 

Differentiating, we have 

dx dy dx ^^ dx ^ 



EXTENDED RULES AND PROOFS 255 

This simplifies to -^=^y? . 
ax q 

We have therefore proved the rule to be true when n is any 

positive number, integral or fractional. 

(c) Let y = x~'^ or yx'^^^ \. Differentiating this product, 

This simplifies to -j-= - mx~'^~^. Thus the rule is proved generally. 

139. I will now prove the rule for exponentials, and incidentally 
I will prove the exponential theorem which is the expansion of a"" in 
a series of powers of x. I assume that there is such an expansion, 

namely, y = a"" = A ■>t Bx + Cx^ + Dx^ -{- Ex^ + q\^., (1) 

and we have to find the values oi A, B, etc. As this is true for all 
values of x, it is true when x = 0, so we have a'^ = A = l. 

y + 8y = a''+^'', and therefore 53/ = «^+^^-a* = a*(a^^- 1). 
Expanding a^^ according to (1), we have 

8y = a' {B.8x+C(8xy + D{8xf + etc.} (2) 

Sothat^ = a*{^ + C.8aj + i).(aa;)2 + etc.}. 

Now let 8x get smaller and smaller without limit, and we have 

&-^^'- (^) 

Apply this to ( 1 ), and we have 

^(l4-Bx + Cx^ + Dx^ + etc.) = B{1+Bx + Cx^ + Dx^ + etc.), 

B + 2Cx-{-SDx^ + etc. = B + mx + BCx^ + BDa^ + etc. 

Equating corresponding terms in this identity, we have 

2C=^2 or C=\B^; 3D==BC or D = iB^etc. 

XT X 1 i> B^x^ B^x^ B^x^ , 

Hence a''=\+Bx + -—--{--j-^ +j— + eUi (4) 

Now let Bx = z, so that ^ = ^> and 

ft« = l + ^+j^ + |y + etc (5) 

1 1 

Let a" be called e, so that -^ log^a =1 or B = log, a. 



256 ELEMENTARY PRACTICAL MATHEMATICS 

With this value of i>, (4) is the usual form of the exponential 
theorem, and (3) becomes, if «/ = a"", 

| = a'log.«. 
Or if // = e*, — = e'. Of course if we use x instead of .:", (5) is 



dx 



e'^=l+a; + ,-y +7-5- +etc (6) 

This is the series given in Art. 95, without proof. 
e can be calculated with any amount of accuracy from (6), by 
taking x=\, for then 

e=H-l+^+^ + |^+etc. 

I have never known a thoughtful young man to be satisfied with 
the usual algebra proof of the exponential theorem [which is the 
modern basis of all calculation of logarithms] ; and I feel sure that if 
he were only allowed to use the calculus idea as I have here done, 
he would have none of his usual feeling of dissatisfaction and 
insecurity. That feeling is mainly due to the complexity of the 
artifices employed. 

140. Lastly, let y = \og{a-{-x). This is the same statement as 
a-{-x = e^. Differentiating with regard to ic, we have 

1 ^dy ^ . ,dy dy 1 

l=e^~ or \={a-\-x)^ or -— = . 

ax ^ dx dx a + x 

141. Exercises in Differentiation. (1) Expand sin a; and cos.^ in 
series of powers of x. 

A ssume 

sinx = A + Bx + Cx^-\-Dx^ + Ex^ + Fx^ + Gx^ + Hx'^-hM-\-Jx^ + et<i., 

where A, B, C, etc., are unknown constants. Differentiating, we get 

cosa; = B + 2Cx + SDx'^ + AEx^ + 5Fx^ + QGx^ + 7Ha^ + Six' + 9Jx^ + etc. 

Differentiating again, we get 

-smx = 2C + QDx+l 2Ex^ + 20Fx^ + 30Gx* + i2Hx^ 

i-5QIx^ + l2Jx'^ + etc. 

Equating like terms in the first and third of these, and recollecting 
that the third is all negative, 

2C= -A, 67>= -B, \2E^ -C, 20F= -D, 306^= -E, 42H= - F, 
56/= -G, 72/= -H, etc. 



EXTENDED RULES AND PROOFS 257 

Now, as sill a: is when x is 0, A is 0. As cos x=l when ;/• is 0, 
^=1, so that 

C=0, ^ = 0, G = 0, 1=0, etc; D= -|, ^'= i^' ^ = 1^' '^=j^ ^^c 

TT . X^ X^ X^ X^ 

Hence sinic = ic-|-^+r-p--r^ + ,-3-- etc., 

2j2 /p4 /^G 

COS ic = 1 - ,--:r + T-j- - r^ + etc. 

II K lA 

These are the series given in Ex. 7, Art. 28, without proof. 

(2) Expand log(l +x) in powers of x. Let 

]og(l+x) = A + Bx + Cx^ + nx^ + Ex^-{-Fx^ + Gx^-^et<i. 

Differentiating both sides, we find 
1 



1 +x 



= B + 2Cx + Wx? + 4 AV + ^Fx^ + %Gx^ + etc. 



But by mere division, we get 

:; = \ - X -^ x'^ - x^ + x^ - 0^ -{■ xf* - x' -\- etc. 

1 +iC 

Equating the coefficients of corresponding terms in these identities, 
we find 

^=1, C=-i i) = i, E=-\, F=l, G=-i, etc. 

Again, when x = 0, log(l +ic) = log 1 =0, so that A is 0, and we 
^^^® \og{l +x) = x - ^x^ + Ix' - Ix^-i Ix^ - etc. 

.1+25 

By writing out log(l - x) and subtracting, and so finding log- 

and then writing for x and by other substitutions, we find series 

which are those used in the rapid calculation of tables of logarithms. 
Many other interesting series may be obtained in the above way, 
but it is more usual to expand any function in series by what is 
called Taylor's theorem. 

142. I will now prove Taylor's theorem. Lagrange was the first 
who introduced rigour into the proof of Taylor; but even to this 
day there is no proof which is perfectly rigorous. I give the usual 
non-rigorous proof. It has to be remembered that an expansion in 
an infinite series has no meaning to the mathematician if it is 
divergent. So in Algebra, sj -\ has no meaning, but Chapters 
XXVIII. to XXXVI. show that when a certain meaning is given to 
J -\ which does not conflict with Algebra, it becomes a very useful 
thing. Mr. Heaviside has made great use of divergent series; he has 

P.M. R 



258 ELEMENTARY PRACTICAL MATHEMATICS 

thereby solved correctly most difficult problems, and it is acknow- 
ledged that his answers are correct, but the mathematician whose 
orthodox methods fail to solve the most elementary of these problems 
is very scornful of the Heaviside methods. 

Proof of Taylm-'s Tlieorem. If a function of « + A be differentiated 
with regard to x, h being supposed constant, we get the same answer 
as if we differentiate with regard to A, x being supposed constant. 
For if the function is called f(i('), where u = x + h, then 
d J., . d J,, . du d J., . du . . 

5s/(«) = a/W><S = a^W as ^ IS 1, 

and this is the same as -^f{'^) because 

d J./ \ d J., . du d r, . du . ^ 

^/W = 5^/(»)x5^ = ^/« as ^ ,s 1. 

If f{x + h) may be expanded in a series of ascending powers of A, 

let it be f(x + h) = A + Bh + ChU Dh^-{- Eh'' + etc., (1) 

where Aj B, C, etc., are functions of x, but they do not contain h. 

^f{x + h) = B + 2Ch + 3Dh^ + 4:Eh'' + etc., (2) 

d J., ,, dA' jdB j„dC .odD ^ ,.v 

3^/(- + A) = ^ + A^, + A^S + A';j^ + etc (3) 

As (2) and (3) are identical, we can equate corresponding terms, 
so that 

j._dA ldB_l^d^A 1 dC_ 1 d^A 

dx' 2 dx~\2^M' ~2. dx~\^d^' 

Also putting A = in (1), we find that A =f(x). If the result of 
differentiating f{x) once, twice, three times, etc., in regard to x be 
written f'{x), f"(x), f"'{x), etc., we have Taylor's theorem 

/(.; + A)=/(.^) + A/(^) + |-/»+ |-/'» + etc (4) 

Having differentiated f{x) once, twice, etc., if we substitute for 
X, let us call the results /'(O), f"{0)y etc. ; if we imagine to be 
substituted for {x) in (4), we have 

m =/(0) + A/(0) + i^/"(0) + '^r(0) + etc (5) 

Observe that we have no longer anything to do with the quantity 
which we call x. We may if we please use any other letter than h 



EXTENDED RULES AND PROOFS 259 

in (5) ; let us use the new letter x, and (5) becomes 

f{x) =/(0) + xf'iO) + 1^/"(0) + 1 J/"'(0) + etc., (6) 

which is called Maclaurin's theorem. 

Ex. 1. Expand {x-\-hY in powers of h by Taylor's theorem. The 
result is called the Binomial Theorem. 

Ex. 2. Expand \og{x + h) in powers of h by Taylor's theorem. 
In the result put x=\ ; this expansion of log (l+h) we obtained in 
another way in Art. 141. 

Ex. 3. Expand sin (x + h) and cos (x + h) in powers of h by 
Taylor's theorem. In the results put x=^0. We have already 
found these answers, which are also the answers of the two follow- 
ing exercises. 

Ex. 4. Expand sin x in powers of x by Maclaurin's theorem. 

Ex. 5. Expand cos x in powers of x by Maclaurin's theorem. 

Ex. 6. Write out the expansions of sin <^ and cos ^ in powers 



of <t>, and show that if i is \/ - 1, expanding e'"^ and e ^'^ according to 
(6) of Art. 28, 

e"^ = cos <^ + i sin <^, e"^'^ = cos<^-^sin<^. 
Ex. 7. Ivaising the expressions in Ex. 6 to the power n, 
show that ^^^g <^ 4- * sin cf>y = cos ncf> + i sin n<f>, 

(cos 4>-i sin </>)" = cos %</> - ^ sin %</>, 

which is Demoivre's theorem. 

The proof of this given in books on Trigonometry is quite easy ; 
it begins with the proof that 

(cos d + isin 6) (cos (^ + i sin <^) = cos {6 -\-<f)-\-i sin (6 + </>), 

which is of course evident from the answer of Ex. 6. 

Ex. 8. Using Demoivre's theorem, find the three cube roots 
of 27. If you like you can say that you are finding the three 
roots of the equation ic^ - 27 = 0. As one of the roots is 3, a: - 3 
must be a factor of x^ - 27. Dividing the equation by a; - 3, we find 

a;2 + 3a; + 9 = 0. 

And the roots of this quadratic are 

x= -|±x/f^= -l•5±2•6^. 

Using Demoivre, however, 27 may be written as 

27 (cos + i sin 0) or 27(cos 27r + isin27r) or 27 (cos 47r + i sin 47r). 



260 ELEMENTARY PRACTICAL MATHEMATICS 

The cube roots of these are 

/ 27r 27r\ 

3 cos or 3 ; 3 ( cos -^ + z sin -^ j or 3 (cos 1 20° + i sin 1 20°) 

or 3(-0-5 + 0-866i) or - 1-5 + 2•6^; 
3 (cos 240° + i sin 240°) or 3( - 0'5 - 0-866i) or -l-5-26i. 
There are only three cube roots ; if we write 27 as 
27 (cos Q-rr + i sin 67r) 
or use higher values, we get these same answers. 

143. Partial Diflferentiation. Hitherto we have been studying 
a function of one variable which we have generally called x. If u is 
a function of two independent variables, say u =f{x, y), we may wish 

to find -r- on the assumption that y is sl constant. I write this 
(-T-V but it may be written merely as -7- if there is a clear under- 
standing that the differentiation is partial. 

Thus in the Heat Conduction problem, Chap. XXXIV., v is the 
temperature at the time / at a point which is at the depth x. Now 
V is a function of x and t. . 

j-g is calculated on the assumption that t is a constant, and -^ 
is calculated on the assumption that x is constant. Thus, if 

In our laboratories we try to make a thing u depend on one 
other thing x only. Thus, in observing the laws of gases : if p 
is the pressure and v the volume and t the temperature (where 
/ = ^° C. + 273) of say 1 lb. of gas, if t is kept constant we have the 

law ^ oc -. If «; is kept constant and the temperature alters, we 

find p <:f.t. After much trial we find that the law pv = Bt is very 
nearly true, B being a known constant, say 96 for air, if v is in 
cubic feet and p is in pounds per sq. foot. Now any one of the 
three, p, v, or t, is a function of the other two, and in fact if any 
values whatsoever be given to two of them, the other can be found. 
Thus , 

P = Bi (1) 



EXTENDED RULES AND PROOFS 261 

We can say that }? is a function of the two independent variables 
t and V. If any particular values whatsoever of t and v be taken 
in (1), we can calculate p. Now take new values, say t + U and 
v-\-^v, where U and ^v are perfectly independent of one another — 
they may be any values — then 

p + 6p = K ^ and bp = R-- . - R ~. 

v + ov v + ov V 

' We see therefore that the change 8p can be calculated if the 
independent changes 8t and 8v are known. 

When all the changes are considered to be smaller and smaller 
without limit, we have an easy way of expressing 8j) in terms of 
8t and 8v. It is , , , , , 

• , ^-&>-&> (^) 

The proof of this is easy. 

The student ought to put this in such words as these: — "The 
whole change in 7; is made up of two parts, (P*) the change that would 
occur in ^ if V did not alter, and (2°'^) the change that would occur in 

p ii t did not alter." The symbol (^\ means, the rate of increase 
of ^ with t when v is constant; the symbol (-^j means, the rate of 
increase of p with v when t is constant. Sometimes the symbols 

^ and ^ are used instead of the brackets for this partial 

ot ov 

differentiation. 

Now the student must understand that (2) is wrong, unless for 
exceedingly small values of 8t and 8v. 

Illustration. Take 1 lb. of air, where i^ is 96. 

Hence (2) is 8p = — . 8t -''^ , h) (3) 

Example. Let ^ = 300, ^^ = 2000, v^U'L These will be found 
to satisfy the law ^ = 96-. Now let t-\-8t = 301 (that is, let 5^ = 1), 
let v + 8v= 14-5 (that is, let ^ = 0'\); it is easy to see that 



262 ELEMENTARY PRACTICAL MATHEMATICS 

^ + 5^ = 1992-82 or 8p= -7-18. This is the true answer, but let 
us use (3), and we find 

. 96 . 2000 . ^._^ 

-^ "^ iXl ^ ^ - ^A'A X *1 = - 7*22, a wrong answer. 

If we had used smaller values for 8t and Sv, we should have had 
an answer from (3) more nearly correct, but (3) gives a correct 
answer only when 8t and Sv are supposed to be smaller and smaller 
without limit. 



CHAPTER XXXIII. 
EXERCISES WITH UNREAL QUANTITIES. 

144. The following exercises ought perhaps to be placed at the 
end of Chapter III., and there are few of them which may not be 
worked by the students who do the other numerical exercises in the 
early part of the book. But it is not necessary that all students 
should work them, and therefore I have placed them here. Until 
men get quite familiar with this kind of work it is quite useless 
for them to proceed further in the book. 

To the end of the book, exercises will be found involving v-T, 
which I usually call i. Such exercises may be neglected by the 
ordinary student. But such students as proceed to the study of 
vibrations of bodies or alternating currents in electricity, to the 
study of telephonic circuits, for example, will find a knowledge of 
these unreal quantities necessary. For a few of the exercises in 
this article a student needs to be able to solve simple simultaneous 
quadratic equations, but such exercises will be worked, presently, in 
a much easier manner. 

Ex. 1. If ^ means V- 1, see that the following values are correct : 
i2=-l, i^= -i^ i4=l, ^5 = 2, i^ = t^= -\^ %> z=zi^= -i^ i^=\. 

Ex. 2. Find \/17 + 30i We know that it is of the form a + bi, 
and we must find a and i, 

17 + 30i = (a + ^>i)2 = a2 - ^>2 + 2a6i. 

Equate the real and the unreal parts separately. 

Hence a^ - b^ = 17, '2ab = 30. We find a = 5-073, b = 2*96, 

so that v/l7-h30i = 5-073 + 2-962, 

Ex. 3. Show that ^/^ = (1 + i)/j2 or -707 + -707?. 

Ex. 4. Show that 1 -^ Vi = (1 - i)/v/2 or -707 - -707^. . 



264 ELEMENTARY PRACTICAL MATHEMATICS 

.. Let it be a-{- hi, and find 

5 4- 3i 
Thus, squaring, = (a + &i)2 = a^ -b^ + '2ahi, 

5 + 3i = 2 («2 - ^2) + 10a^> + { 4ft6 - 5 (a^ _ J2) } i^ 
Hence, equating real and unreal parts, 

2(ft2_62)+iOa6 = 5, 
4ft6-5(ft2_Z>2) = 3, 

From these we find a = 0-6747, ft = 0-7927, so that the answer is 

0-6747 +0-7927e. 
Ex. 6. Extract the square root of 5 + 4i. Ans. 2*389 +0-8365i. 

Ex. 7. Extract the cube root of - 2-35 + l-96i. 

Anii. 0-9959 + 1-Oo67i. 

Exercises like the above are much more easily worked in the 
following way. 

145. The following way is easy, but only to a student who can 
readily find the sine, cosine, and tangent of any angle. Let him 
refer again to Art. 34 for the definition of these. 

In Fig. 46, if OF is 1 inch, the 
length of FB, in inches is the sine of 
the angle FOFi, and the length of OR 
is the cosine of the angle. Let FFi, be 
called y and OFi be called «, then y is 
the sine and x is the cosine of the 
angle FOF^ and y -r x is the tangent 
Fig. 46. ^ ^f the angle. 

To find the sine, etc., of any angle. 
Draw XOK and YOS (Fig. 47), both produced, at right angles to 
one another. 

Let the angle be XOF' or XOF" or XOF" or XOF"" . 
The angle is in every case measured from OX in the anti-clockwise 
direction. XOF' is said to be in the first quadrant; it is between 
and 90°. XOF" is said to be in the second quadrant; it is 
between 90° and 180°. XOF" is said to be in the third quadrant; 
it is between 180° and 270°. XOF"" is in the fourth quadrant ; it is 
between 270° and 360°. 

In every case make OF' or OF' or OF" or OF"" one inch 
long. The vertical distance of F from XOK is called y or the sine 




EXERCISES WITH UNREAL QUANTITIES 265 

of the angle ; it is positive if P is above XOK, and it is negative if P 
is belmu XOK. The horizontal distance of P from YOS is called x or 
the cosine of the angle ; it is positive if P is to the right of YOS, 
and it is negative if P is to the left of YOS, and y^xis the tangent 
of the angle. 

Y 




The student ought to practise finding the sine, cosine, and tangent 
of angles whose magnitudes are between 0" and 360°. First he finds 
by making a sketch in which of the four quadrants the angle is. 

It helps greatly to remember that if A is an angle greater than 90° 
and less than 1 80°, so that it is in the second quadrant, 
sin ^ = sin (180°-^), 
cos ^ = - cos (180° - A), 
tan^= -tan (180°-^). 
Thus sin 160° = sin 20° = 0-3420, 

cos 160°=- cos 20° = - 0-9397, 
tan 160° = - tan 20° = - 0-3640. 
Again, in the third quadrant, 

sin^= -sin (^-180°), 
cos A= - cos (A - 180°), 
tan ^ = tan (^-180°). 
Thus sin 210° = - sin 30° = - 0-5000, 

cos 210°= -cos 30°= -0-8660, 
tan 210° = tan 30° = 0-5774. 



266 ELEMENTARY PRACTICAL MATHEMATICS 

Again, in the fourth quadrant, 

sin^= -sin (360°-^), 
cos ^ = cos (360°-^), 
tan^= -tan (360°-^). 
Thus sin 320" = - sin 40° = - 0-6428, 

cos 320° = cos 40° = 0-7660, 
tan 320° = - tan 40° = - 0-8391. 
Again, for negative angles we must remember that 
sin(-^)= -sin^, 
cos( -^) = cos^, 
tan( - A)= - tan^. 
Let us tabulate all the above answers : 



Angle. 


Sine. 


Cosine. 


Tangent. 


20 
160 
210 
320 


0-3420 

0-3420 

-0-5000 

-0-6428 


0-9397 
-0-9397 
-0-8660 

0-7660 


0-3640 
-0-3640 

0-5774 
-0-8391 


-20 
-160 
-210 
-320 


-0-3420 

^0-3420 

0-5000 

0-6428 


0-9397 
-0-9397 
-0-8660 

0-7660 


- 0-3640 
0-3640 

-0-5774 
0-8391 



I am afraid that there is no help for it ; if a student is to do the 
following exercises with accuracy he must practise finding the sines, 
cosines, and tangents of angles so much that he is perfectly certain 
as to the + or - sign and the number in each case. Inaccuracy as 
to + or - is fatal in this work. 



146. Demoivre's Theorem. It is easily proved in trigonometry 
(see also Exs. 6 and 7, Art. 142) that the t}}'^ power of 

?-(cos</) + esin(^) is r"(cos 7i<^ + «' sin ti^), 
where r is always positive, <^ is any angle, and n any number, positive 
or negative. Of course ^ is usually expressed in radians, but I often 
find it convenient to write it in degrees. To save time and for 
other good reasons, I often write r(^) or r[^], and indeed sometimes 
r^, instead of ?-(cos <^ + i sin <^). 

Again, 
r(cos </) + i sin <^) X r^(cos 4>i-\-i sin<^i) = ri\ [cos (</> + </)j) -i- ^ sin (<^ + <^i)} 



EXERCISES WITH UNREAL QUANTITIES 267 

or r[4i]^r^[h] = r7\[i> + <t>^], 

V 

?•(cos<^ + ^sill<^) ^rj(cos</)i 4-isin(^j) = — {cos (t^ - </)^) + isiii(<^ -^\)} 

or »-M-^iM = f[<^-<^i]. 

Again, it is known that we get consistent answers if we use e** 
instead of cos <^ + i sin <^ in all calculations. Again, e~^* is the same 
as cos 4>-i sin <^. 

In fact r(cos <^ + i sin ^) is, algebraically, the same as 

re'^ or a -{-hi if rcos<^ = a and rsin</) = 6. 



EXERCISES. 

1. Express 1-452 [46° '7] in the form a + hi; 
that is, 1-452 (cos 46°•7 + ^ sin 46°-7) 

or 1-452(0-6858 + 0-72780 or 0-9958 + 1-0567*. 

2. Express 5 + 4^ in the form r[<^] or r (cos ^ + 1 sin <^). 

Here rcos</> = 5, rsin^ — 4. By division, therefore, tau<^ = 0-8, and 
from the tables ^ = 38° 67. Also ^2 = 5^ + 4^ or r = 6-403. 
The answer is then 6-403[38°-67]. 

3. Extract the square root of 0*9958 + r0567i. It is easy to show 
that the given expression is 1 '452 [46° -7], and the square root of this is 

l-452*[46°-7 + 2] or l-205[23°-35], and it is easy to put this in the form 
1-107 + 0-47761. 

4. Raise 0*9958 +l-0567i to the third power. As in the last case, 
convert the given expression to 1*452 [46° *7], and the answer is 

l-4523[46''-7x3] or 3-062[140°] or -2-35 + l-96i. 



5. Find VTT + SOi. Here 17 + 30^ may be converted into 34-48 [60° -45], 
and its square root is 5-872 [30°-23], which may be converted into 

5-073 + 2-961. 

6. Find 'sji. Now i is 1 (cos 90° + i sin 90°) or 1[90°], and its square 
root is 1[45°] or cos 45° + 2 sin 45° or 0-707 + 0-707i. 

7. Findl+^/^T This is 

r^ or r^[90°x(-i)] or l[-45°] or 0-707 - 0-707i'. 

V5-f-3? 
:. Converting separately the numerator 
2 — 5i 

and denominator of the given fraction, 

5 + 3*-_ 5-832[30°-96] _ i-ogsfgr-Hl 
2-5i-5-386.{-68°-18]-^ ^^-^^^^ ^*-'' 

and the square root of this is 1041 [49°-57] or 0-6747 +0-7927i. 



268 ELEMENTARY PRACTICAL MATHEMATICS 

9. Extract the square root of 5 + 4i. This can be expressed as 
6-403[38°-67], and its square root is 2-53[19°-34] or 2-389 +0-8365i. 

10. Extract the cube root of - 2*35 + 1 "961. This may be put in the 
form 3 "06 [140°] ; its cube root is 1*452 [46° '7], and this is converted into 
0-9958 + 1 -05671. 

11. Expand e"^ according to the rule of Ex. 5, Art. 28. Now expand 
cos (f) and sin </> according to the rules of Ex. 6, Art. 28. Show that 

e^^ = cos (f) + t sin (f) = [<^]. 

In the same way show that e~^^ = cos <^ — i sin <j> — \_ — <^]. 

147. As in Art. 119, when r[(^] operates upon a%\\\qt (that is, 
when a^mqt is multiplied by ^' [</>]), it converts it to 

ar sin {qt + (f>). 

Ex. 1. Operate with 5 + 4i upon 10 sin ^^. Here we convert 
5 + 4z into 6*403 [38° -6 7], so that the answer is 

64*03 sin (^^ + 38"-67). 

The student ought to see that this agrees with the rule of 
Art. 119. 

Ex. 2. The voltage applied at the sending end of a long 
telephone line being Vq sin^^, the current entering the line is 



V: 



's -\-kQi 

VqSIIi qt, 



r + Iqi 

where, per unit length of line, r is resistance, / is inductance, 5 is 
leakance, and k is permittance or capacity. 

If, per mile, r = 6 ohms, Z = 0-003 henry, A; = 5xl0~^ farad (or 
as it is usually stated 0-005 microfarad), s = 3xl0~'^ mho; if 
q = 6000, find the entering current. 

Ans. The operator is 10""^^/- — -- ., and is found to be 

V 6 + 1 8z 

0*00126[6°*35], 

so that the entering current is 0*001 26«;o sin (2^ + 6''*35). 

148. In using 6 or -r, or its equivalent qi, as if it were an 

ordinary algebraic quantity, it is to be observed that it is an 
operator upon a quantity which is a function of the time like 
asiwqt. I have nowhere operated upon the product of two such 
functions. 

Again, I have said that a sin {qt + e) may be represented by a 



EXERCISES WITH UNREAL QUANTITIES 269 

vector when we add such functions, but I have been careful not to 
speak of the product of two such functions as the product of two 
vectors. The fact is, the name " product of two vectors " may get 
all sorts of meanings. Two of these are very useful in Vector 
Algebra, and neither of them has anything to do with the pro- 
duct of two algebraic expressions like a sin (qt + e). (See Chap. 
XXXVII.) 

Operators like a + U and a-\- pi may be added like vectors, and 
the sum is a + a + (b + I3)i, or they may be multiplied algebraically, 
and their product is truly aa - b/3 + {afS + ab)i, but this product of 
operators need not be called a vector product. We have already 
two very important things that we call vector products; it is not 
wise to have a third and very different thing with the same flame. 
But there is no harm in speaking of the product or quotient of 
operators. 

149. coshcK means ^(e' + e~''), 

sinh X means J(e* - e~^). 
tanhic means sinh a; -r cosh j.* 

These are called hyperbolic functions, because they originated in 
a study of the hyperbola just as sin«, coscc, and tana; are called 
circular functions, because they were first derived from the study 
of the circle. 

There are some cases in which it is a pity that men know the 
history of a subject or the origin of a word ; they get to think 
that this kind of knowledge is the only essential knowledge. I 
am sometimes thankful for the oblivion that hides from us the 
wonderful romance that must have been once attached to the in- 
vention of every common idea or every common word that we use. 

* The student does not need to know the following formulae for doing the 
exercises in this book ; but he will need them if he pursues these subjects, and 
they are very easy to prove from the above definitions. 

sinh (a + ft) = sinh a . cosh b f cosh a . sinh 6. 
cosh (a + b) = cosh a . cosh h + sinh a . sinh ?/. 
sinh {a-h) = sinh a . cosh h - cosh a . sinh h, 
cosh (a - 6) = cosh a . cosh h - sinh a . sinh 6. 

sinh 2a = 2 sinh a . cosh a. 

cosh 2a = cosh^a + sinh^a = 2 cosh^a -1 = 2 sinh^a + 1 . 

cosh^a- sinh''a = l. 



270 ELEMENTARY PKACTICAL MATHEMATICS 

I wish we could drop such terms as hyperbolic functions ; they have 
served their purpose and now do harm, for teachers will insist on 
deriving their properties from the hyperbola, and this is foolishness ; 
the definitions are perfectly easy to deal with as I have given 
them. 



EXERCISES. 

1. Show that the differential coefficient of cosh aa; = asinhax. 

2. Show that the differential coefficient of sinh a^=a cosh ax. 

3. Show that 

cosh 1-013 = 1-5562, sinh 1 '013 = 1-1989, tanh 1-013 = 0-7704. 

4. What are the values of cosh a, sinh a, and tanh a when a is 
small ? 



l+a + ~+-g+etc. and e-"- = l-a + Y 



6 



+ etc., 



As e" 

cosha = ^(2 + a2 + etc.) = l + ^«2_j.etc., sinh a = |(2a + ^a2) = a + ^a^ + etc. 
Calculate the following values to four significant figures : 



a 


cosh a 


sinh a 


tanh a 




0-01 

0-10 


1 

1 

1-005 




01 
01002 




o-oi 

09967 



5. Show that we may take cosha = sinha=^e" and tanha=l if we 
are only using four significant figures, and if a is greater than 7. 

6. If X is a + bi, find e". This is e«+w = e'V^ = e«[6]. Express this in 
the following cases, stating 6 in degrees and also converting the answer 
into the a + Bi form : 



X 


^ 


0-01+0 OH 


1 -01 [0°-573] = 1 -0099 + 0-OlOH 


0-1+0-H 


1- 105 [5° -73] = 1-0995 + 0-1 1017* 


0-3 + 3i 


l-350[17°-19]=l-2898 + 0-3989i 


0-6 + 0-6i 


1 -823 [34° -38] = 1 -5045 + 1 -0285* 


1+i 


2-7183[57°-30] = 1 -4684 + 2 -28741 


1-5+1-5? 


4-482[85°-95] = 0-3169 + 4-47121 


2 + 2i 


7-389[114°-60]= -3-0753 + 6-7188« 


3 + 3i 


20-086[171°-90]= -19-887 + 2-8300i 


5 + 5i 


• 148-41 [286°-50] = 42-14 - 142-28i 



EXERCISES WITH UNREAL QUANTITIES 271 

7. Find e~* in the following cases, li x = a + bi,e-'' = e~"^e~^^ = e~"[-b]: 



X 


c-^ 


0-01 + 0-Oli 


0-9901 [ - 0° -573] = -9900 -0•0099^• 


01+OH 


0-9053[ - 5°-73] = 0-9008 - 0-09026i 


0-3 + -Si 


0-7407L- 17°-19] = 0-7077-0-2189i 


0-6 + -61 


0-5486 [ - 34° -38] = 045275 - -309791 


l + i 


0-3679[ - 57°'30] = 0-19874 -0-3096* 


1-5 + 1 -51 


0-2231 [ - 85°-94] = 0-01577 - 0-22256* 


2-+2i 


0-1353[- 114°-60]== -005631 -012303* 


3 + 3i 


0-0498[- 171°-90]= -0-04931 -0-07016*" 


5 + 5* 


0-00674[ - 286° -50] = 0-001914 + 0-006462* 



8. Combining the answers to (6) and (7), find cosh^ and sinh;r for 
the following values of x : 



X 


coshz 


siuhx 


0-01 + 0-01* 


1+0-0001* = 


= 1-0(0°) 


0-01 + 0-01* = 


= 001414(45°) 


01+0-1* 


10001 5+0-009955* = 


= 1-0(0° -56) 


0-09935 + 0-1002* = 


= 0-1411 (45° -2) 


0-3 + 0-3* 


0-9988 + 0-09* = 


= 1-03 (5° -2) 


0-2911 + 0-3089* = 


= 0-4244 (46° -7) 


0-6 + 0-6i 


0-9786 + 0-35935* = 


= 1-042 (20° -2) 


0-5259 + 0-6692* = 


= 0-8511 (51°-8) 


1 + * 


0-8336 + 0-9889* = 


:= 1-293 (49° -9) 


0-6349 + 1-2985* = 


= 1-445 (63° -9) 


1-5 + 1-5* 


0-16634 + 2-1243* = 


=2-13(85°-5) 


0-15057 + 2-3469* = 


= 2-352(86°-3) 


2 + 2*- 


-1-5658 + 3-2979* = 


= 3-65(115°-4) 


-1-5095 + 3-4209* = 


= 3 -74 (113° -8) 


3 + 3* 


-9-968 + 1-3799* = 


= 10-06(172°-]) 


-9-919 + 1-4501* = 


= 10-02(171°-1) 


5 + 5* 


21071 -71-137* = 


= 74 -19(286° -5) 


21 -069 -71 143* = 


= 74-19 (286° -5) 



Tables of cosh a and sinha are easily procurable. The student who 
wishes to calculate cosh x or sinh x, where .v = a + bi, may find the following 
formulae convenient. Let him first prove them to be correct : 

cosh (a + bi) = cosh a . cos b + i sinh a . sin 6, 
sinh (a + bi) = sinh a . cos b + i cosh a . sin b. 

Prof. Kennelly has published a very complete table of cosh a*, sinh^', 
and tanh^, where A'=r[45°J or a + ai. 

9. The following values of r, I, s, and k are values per mile for certain 
known submarine and telephone lines. It is known that in telephone 
lines if a pure musical note of a frequency or pitch of about 800 per 
second is transmitted well, ordinary speech will also be transmitted well. 
I therefore take q = 5000 in telephone lines, q being 27r times the frequency. 
Also in submarine telegraphy if a simple harmonic current of frequency 
about 9 is transmitted well, ordinary signalling at the rate of .160 letters 
per minute will be effectively written by the ordinary recorder. Therefore, 
for submarine cable work I use g' = 60. 

Telephone lines differ greatly, and therefore the results of experiments 
are usually given in terms of what is called the " standard telephone cable," 



272 ELEMENTARY PRACTICAL MATHEMATICS 



about thirty miles of which is about the limiting length along which 
speech may be transmitted so as to be heard articulately when the 
ordinary transmitters and receivers are used. 

Calculate n = h-\-qi= s/(r + qli) {s + qki). 

Calculate Zq = s/{r + qli) -^ (s + qki). 

I often, instead of using Zq, call it r/n, really meaning ^ ^^ . The 
answers are given in the following table ; ^ 







^ 




1 — 1 








!>. 




00 








CO 


o 


^ 






5 ' 





•P 


. £"■ 


^IS 


•j2 >o 


SS 


.^^ 


-^k 


?!? 




Oi 1 


»■ 1 


00 1 


Tfl 1 




o 


■^ r^ 


8^^ 


CO J_, 


o J_, 


^^ 


o 




^,' »o 


i F^ 






1 35 


1 -H 


^ a> 


1 Oi 


1 ^ 




^ lO 


osoo 


1 '^ 


"p-r 


00 CC 




05 II 


,1h II 


Oi II 


o II 


^ II 




■<* 


£e 


00 


o 


^ 






,__, 


,— , 




■cS 






o 


(M 


,— , 


00 






Oi 


•^CO 


Oi 


St! S"' 


U 


to 


ife 


Ik 


11 


|s 


+ 


O CO 


OS 


9-^ 


o ■* 


-s 
s 


<^i^ 

ii 


og^ 


+ o 


oo8 


o 


ooo 


8 " 


o '^ 




6 


6 


6 


6 


6 


r! 




? 


to 


tc 






o 


. 1 
o 


o 


s 


o 


S 








X 












lO 






« 


to 


<o 


«c 


5C 




o 


1 

o 


1 

o 


o 


O 


• 












^1 


X 

LO 


X 


X 


X 


X 


c9 


9 


g 


Q 


t-- 


o 




o 


• 


^ 


9 


»* 






o 


6 


o 


6 


. 




05 


cc 


o 




o 












~i 'n 


o 


8 


8 


8 


o 


^ 




6 


o 


6 




<a 






t-- 




00 


i 


00 


00 


Oi 


(M 


00 


00 




Ol 




OJ 




^ 


^ 


^ 


^ 


^ 


w 


X 


g 


8 


g 


tc 




>3t 


o 


lO 


O 






t 


£ ■ 


.£ ' 


• 


■a 






'K 


■^0, 


0) 


1 




^ 


a 


a 


1^ 


a> 






S-i "-* 


f, ..- 




rs 




2 


© ■— ' 


S'- 


1 


^ 




n 


Ph <» 


P4« 




Sc 


Sg 


03 


© 




1 


o o 


c o 


c 


.2 




ox 


"-^ 


o 








&4 


^ 


X 


e3 




1 


G^ 


G-S 


s^ 


C 




1^ 


|i^ 


0) 

13 


:3 




OQ 


o 


O 


H 


cc 



^ 






QT) 


'Ti 


(M 


a 


, 


a 


;h 


q 


-II 


^■T 


■^ 




(M 


ft; 


X 


'^ 


W 


srj 


O 


QJ 




fl 


W 


•—m 


o 


-^ 


:S 




,£3 


Tl 


-t-3 


C-) 




a> 


i? 


^ 


'Si 


;-i 


1 


cS 


OS 


^ 


-< 


r^ 


;-i 


^ 


^ 


t4 


33 


a> 


;-i 


T1 


<D 


b 


(* 


o 


^ 


m 


c 




ce 


^ 




-u 








G 


Ci) 




_ r-] 


crj 


-t-> 


0) 


<D 


c 


(^ 


^ 


P. 


«3 


s 


(V 


o 


e 


o 



EXERCISES WITH UNREAL QUANTITIES 273 

10. In this table I give distances L miles of each of the abov^e lines. 
Calculate cosh Ln^ sinh Z??, and tan Ln in each case : 



Line. 


L miles. 


cosh Ln 


siuh Ln 


taiih Ln 


A 


30 


ll-685[180°-5] 


ll-645[180°-5] 


0-9965 [0°] 


»> 


5 


1025 [15° -6] 


0-7437 [50° -3] 


0-7256[34''-7] 


B 


150 


3 -041 [259°] 


3 -19 [259° -5] 


l-049[0°-5] 


>> 


25 


0-7911 [15° -6] 


0-7526[7-2°-5] 


0-9513[56°-9] 


C 


200 


0-995[339°-l] 


0-8421 [304° -2] 


0-8457[325°-l] 


j» 


33 


0-605 [7° -3] 


0-8077[85°-9] 


1-335 [78' -6] 


D 


60 


4-993[193°-6] 


4-904[194°-l] 


0-9822[0°-5] 


j> 


10 


0-9325[13°-l] 


0-6642[59°-9] 


0-7123 [46° -8] 


E 


2432 


952820[828°] 


952820[8-28°] 


l-0000[0] 


>j 


392 


5-154[134°] 


5-159[133°] 


1 -0008 [1° -07] 



11. For the standard cable A above, find the following answers 
(^ = 5000): 



L 


sinh Ln 


cosh Ln 


tanh Ln 


1 

5 

10 

20 

30 


0-1483[45°-4] 
0-7437 [50° -3] 
l-525[65°-9] 
4-097[119°-4] 
ll-65[180°-5] 


0-9998[0°-6] 
1-025 [15° -6] 
1-348 [53° -7] 
4-034[120°-9] 
11 -69 [180° -5] 


0-1484[44°-8] 
0-726[34°-7] 
1-131 [12° -2] 
l-016[-l°-5] 
0-9965[0°-0] 



12. For the standard cable A above, find the following answers 



:^ooo 

5000 

7000 

10000 



541-5(1-1) 
419-5(1-1) 
354-5(1-1) 
296-6(1-1) 



0-08122(1 + z) 
0-10486(1 +*) 
0-12408(1+*) 
0-1483(1 +i) 



cosh 40n 



12-875[186°-15] 
33 -156 [240° -33] 
7 1-55 [284° -40] 
188 -5 [339° -9] 



cosh 60n 



65 -5 [280° -23] 
270[360°-51] 
855 [426° -77] 

3675 [509° -9] 



cosh 80n 



332[372°] 

2194[480°-45] 

10246 [568° -71] 

7 1130 [679° -97] 



These results are important when comparing the telephonic currents 
received at the distances of 40, 60, and 80 miles. 



P.M. 



CHAPTER XXXIV. 
FUNDAMENTAL EQUATIONS. 



150. There are certain fundamental equations concerning the 
conduction of heat, the flow of electricity in telephone or submarine 
telegraph conductors, the flow of water or other fluid, the trans- 
mission of electric or electromagnetic conditions, the diff"usion of 
fluids, etc., which all take the same mathematical shape. It results 
that when a man has worked a problem in any of these sciences, he 
has worked an analogous problem in all the other sciences. In my 
Calculus I have described some of the methods by which problems 
are solved ; I shall here find the fundamental equation when heat is 
flowing in one direction. 

151. Conduction of Heat. If material supposed to be uniform 
has a plane face AB (Fig. 48). If at the point F, which is at the 

distance x from AB, the temperature is v, and we 
imagine the temperature the same at all points in 
the same plane as F parallel to AB (that is, we 
are only considering flow of heat at right angles 



Q 



to the plane AB), and if -^ is the gradient of rise 

of temperature per cm. at F, then -kdv/dx is the 
amount of heat flowing per second per square cm. 
of area like FQ in the direction of increasing x. This 
is really a definition of what we mean by k, the 
conductivity of the material. I shall imagine k to 
be constant, k is the heat which flows per second 
per square cm. when the temperature gradient is 1. 
Let us imagine FQ exactly a square cm. in area, and 

dv 

FT or ^aS^ is Bx,. The heat current per second, c= -k-.-, flows 

ax 

into the block FQTS through the face FQ ; how much flows out at 



Fio. 48. 



FUNDAMENTAL EQUATIONS 



275 



dc 
the face TSI Call it c + Sc or c + hx-^. Then the amount of heat 

^^ dc 

in the block diminishes by the amount ^^-r- per second, or the 

increase per second is 



Bx.k 



dx' 



Now the weight of the block is I x Sic x w, if w is the weight per 
cubic cm., and if s is the specific heat of the material or the heat 
required to raise unit weight one degree in temperature, then if t is 
the time in seconds, 



8x .w. 



dt 



also measures the rate per second at which the heat in the block 



increases. Hence 



8x.k 



dh 



Bx.w. s-jj 
dt 



or 



dx^ 

d^v _ ws dv 

dx^ k dt' 



(1) 



ws d 



We may call this j^ = '^^^'j using n^ for -r - j., the fundamental 
equation of which I spoke. 

152. Telephone or Telegraph Cable. Imagine a metallic con- 
ductor insulated from an infinitely perfect return conductor (of 
no resistance) which we call earth, and which is everywhere at 
potential ; at a place A (Fig. 49), which is at the distance x from 
the sending end, the potential is v and the current is c ; at the place 





















B 











^J. 



I J 

Fig. 49. 



B, which is at the distance x-\-8x from the sending end, the potential 
is V + 8v and the current is c + 8c. I assume c to be in the direction 
of the arrow. The current at ^ is 



^ _ldv 
r dx 



where r is the resistance per unit length of conductor. The student 
must not proceed till he sees clearly this simple thing, that the 



276 ELEMENTARY PRACTICAL MATHEMATICS 

current in any conductor is the voltage gradient divided by r. The 

dc 1 dr'V 

current at ^ is c + 8c or c + ^x --, so that 6c = -h:- -j—. 

ox r dx" 

But if there is leakance s mhos per unit length of conductor, 
there is a current leaking away sideways whose amount is v .s . Sx. 
Hence the rate at which the quantity of electricity in the space 
between A and B is increasing per second is 

. I dh 

ox ~ -^-B -v.s .ox. 
r dx^ 

Now AB is one coating of a condenser whose capacity is h . 8x, if 
k is the capacity per unit length ; the charge or quantity of electricity 

in the space AB is k. 8x . v, and it is increasing at the rate k . 6:^^, 

ill 

if t is time. We have two expressions for the same thing, and so 
^ 1 d^v » 1 Si dv 

ox- -^j-^-V .S.OX = k.OX-rr 

r dx^ dt 

or ^^ = T{s + ke)v, 

if we use 6 as meaning -y-. 

But if the conductor has self-induction I per unit length (see 
Art. 128), we use r + W instead of r, and so we have the fundamental 
equation /72., 

~-{r+ie)(s+ke)v (1) 

dH 
We may call this -v-g = n'^v. 

Notice that this is the same equation as we had for heat, only 
that in the case of heat we had no /, and as there was no side leak, 
we had no s. In studying heat conduction along a bar, we should 
give s a value, as heat would leave the rod sidewise. 

In the following work v is voltage in the electrical problem and 
c is electrical current But v may mean temperature in the heat 
problem, and c will then mean the current of heat per sq. cm. per 
second. In any problem the student must not be worried about n 
being a function of 6. 

When dealing with functions like sing/, he is already familiar 
with such a quantity as n, because 6 is then merely qi. 



CHAPTER XXXV. 

TELEPHONE AND TELEGRAPH PROBLEMS. 

153. If in (1) of Arts. 142 or 143, n were a real quantity, the 

solution would be v = Pe'^ + Qe~"^, (1) 

where F and Q are arbitrary constants. But as n involves -^, instead 

of taking P and Q as mere constants, they ought to be regarded as 
functions of the time. In cases of x reaching to infinity P must 
be 0, and we have for an infinitely long line 

v = e-''^Q, (2) 

where x = 0, v = Q,so that Q is a function of the time expressing how 
V varies at a: = 0, and it is evident that, to find the value of v for any 
value of X, we have to perform the operation indicated by e""^ upon 
Q. In the cases which I shall consider, Q and P will be of the type 
sin qt, and the exercises already given show how easily practical 
problems may be worked. Remember that in differentiating with 
regard to x, Q is a, constant. 

When X is limited, P in (1) has a value, and it is found more 
convenient to use what is equivalent to (1), 

v = A cosh nx + B sinh nx. (3)* 

As c = - - J- in the electrical case, 

r ax 

c= — (^ sinh riic + jB cosh ?!«), (4) 

where r stands for r + 19 or r + Iqi. Prof. Kennelly employs z^ for 
r + lqi 

n 

(3) and (4) enable us to work all sorts of problems. 

*Co8h7ix is an operator upon A which is a function of the time, so it might 
be more logical to write coshnxxA, but there is less risk of algebraical 
mistakes if we write it A cosh nx. 



278 ELEMENTARY PRACTICAL MATHEMATICS 

154. Suppose at the sending end of a telephone line v^v^sinqty 
and the line is infinite in length. I shall choose the formula (2) to 

work with. Here c = ~ Qe''"^, where r stands for r + Iqi. We can 

evaluate n when we know r, I, s, and k. Q is evidently v^ sin qt, but 
I shall consider sin^^ as being everywhere understood, and I need 
not write it. I shall say that Q is v^. In fact, everywhere 

operating upon sin qt^ and n = \/(r-\-lqi){s + kqi). 

When r, I, s, k, and q are given numerically"^ it is easy to find 
n in the shape h + gi. Now g-'^+i'''^ = e-^[ - gx\ 

If at the sending end v (or the value of v when x is 0) is v^ sin qt, 
the value of v anywhere else is 

V = v^e~^ sin {qt - gx). 

That is, V attenuates because of the multiplier e~^ and it lags by 
the amount gx. When the lag amounts to 27r we may, if we like, say 
that this gives a complete wave. If A is the length of the wave, 
^A = 27r or A = 27r/^. 

It is to be observed that c attenuates and increases its lag in 

exactly the same way as v.. In fact c = ^ — y-. or vjz^^. 

Ex. L In what is called the Standard telephone cable, r = 88 
ohms per mile, Z = 0, 6 = 0, A; = 0*05 x 10"^ farad [this is usually called 
05 microfarad, a microfarad being the millionth of a farad]. Find 
n either from the formula (1) or from (2). Also find {r-\-lqi)ln, 

* I hope that the student does not mind doing algebraically what he has 
hitherto done, in the exercises of Chap. XXXIII. , upon numbers. The lazy 
student will take my answer (2) here to be correct. 

n = h + gi=isl{r + lqi){s + kqi). (1) 

h'^-g^ + 2,hgi =rs- Ikq^ + i {rkq + slq). 
Then h^-g^=r8-lk(f 

and 2A,gr = g'(rA; + s/). 

Solving these equations for 7i and g, we find that 

VfVVFWFiH?^ <^' 

is the value of h if the minus sign is taken, and it is the value of g if the plus 
sign is taken. 

It is easy to show that when ^~ is large compared with 1 and if s is 0, it is 
very nearly true that ^ 

and g = q\lkl. (3) 

This is a good example of the fact that when a is small, \/l + a = 1 + Ja. 



-iVf 



TELEPHONE AND TELEGRAPH PROBLEMS 279 



which I usually call rjn or z^^. Give three answers for the four 
values of q, 3000, 5000, 7000, 10000. Using formula (1), n = s/rJcqi. 



1 


n = h-\-qi 


J °" ^« 


3000 

5000 

7000 

10000 


008124 + 0'08122i = 0-1149[45°] 
0-10486 + 0-10486i = 0-1483[45°] 
0-12408 + 0-12408i = 0-1754[45°] 
0-1483 +01483i =0-2097 [45°] 


541-5(1 -^•) = 766 [-45°] 
419-5(1 -i) = 593[- 45°] 
354-5(1-*) = 501 [-45°] 
296-6(1 -i)=419[-4oT 



Ex. 2. If Vq= 1, find v and c at the end of 30 miles for each of 
the above values of q. Taking the case of g = 5000, A = ^ = 0-10486, 
and as v anywhere is ?; = e~°^^'^*^sin(g^/- 0-10486a:), we have the 
answer given in the table. For the other values of q we have 
similar expressions. I give the lags in degrees instead of radians. 

The current c anywhere is ?; x - or vjz^. 



9 


V 


c 


3000 

5000 

7000 

10000 


0-08744sin(g<-139°-6) 
0-04303 sin (5^-180° -2) 
0-02418 sin (g<- 213° -27) 
0-01169sin(5<-255°-0) 


0-0001142sin(^r-94°-6) 
0-00007-253 sin (g^- 135° -5) 
0-0000482 sin (g«- 168° -3) 
0-000028 sin (g«- 210°) 



Note that c leads v in every case by 45°. Note the gi*eater 
attenuation and lag of the higher frequencies. 

Ex. 3. If Vq=1^ find v at the following distances x miles from 
the sending end when q = 5000. The answers are : 



e~ 


'"""^smiqt-O'lOiSQx). 


1 


0-9004 sin (9<- 6° -008) 


5 


0-5920sin(g«-30°-04) 


10 


0-3504 sin (g^- 60° -08) 


15 


0-2074 sin ((7<-90°-12) 


20 


0-1227 sin (g<-120°-16) 


25 


0-07269 sin (g«- 150° -2) 


30 


0-04303 sin (g^-180°-2) 


35 


0-02548 sin (g«- 210° -24) 


40 


0-01508 sin (g<- 240-3) 



Plot these values of v and x, and note how v attenuates. 
A the wave length is 60 miles. 



280 ELEMENTARY PRACTICAL MATHEMATICS 



155. It was discoA^ered by 'Mr. Oliver Heaviside that the ordinary 
telephone or telegraph line whose / is usually so small as to be 
taken as 0, will attenuate current much less if / is large. Try this 
in the following example. As in the Standard cable take, per mile, 
r=88, k = 0-ObxlO-\ s = 0, but / = 0-1 henry. 

Take ^ = 5000. Using either (1) or (2) or (3) of the note (Art. 
154), we find 71 = 0-0311 +0-3536i = 0-355[84:°-97]. 

Hence, if ^o = 1, ^ = e-°°=^'^" sin {qt - 0-3536a;). 

r 4- Iqi _ 8 8 + bOOi __ 507-7 [80°-02] 
n ~0-355[8r-97] 



Note that ^^ or 



1430[-4°-95]. 



That is, the current everywhere is v divided by 1430, and it 
leads V only by about 5° ; that is, v and c are nearly in synchronism. 



X 


V 


1 


0-9694 sin (g« - 


-20° -26) 


5 


0S5QI sin {qt ' 


-101°-3) 


10 


•73-26 sin (g< - 


-202° -6) 


15 


0-6272 sin (g« - 


-303° -9) 


20 


0-5369 sin (g^ 


-405° -2) 


25 


-4596 sin (5< - 


-506° -5) 


30 


-3933 sin (5< - 


-607° -8) 


35 


0-3367 sin (g« - 


-709°-l) 


40 


0-28828in(5«- 


-8 10° -4) 



The effect of the introduction of Self-induction is obvious if 
we compare these numbers with those of the last example, and 
especially if we show the results in two curves. 

This method is now carried out largely in practice. 

In finding n in the above example a student can greatly shorten 
the work if he uses the formulae given in the note of Art. 154. 

When ^ is greater than 4, we have, approximately, 

h = ~J- and g^qjkl. 

In fact currents of all frequencies are attenuated to the same 
extent, and because of g being nearly a multiple of q, we have 
almost no distortion of complex currents. 

Ex. 1. An infinite line, the standard cable A of Art. 149. 
Currents whose frequencies are 5000 ^27r and 10000 -=-27r are sent 
into the line. Prove that the current of lower frequency is halved 
and lags 39° in every 6-6 miles ; the current of higher frequency is 
halved and lags 39° in every 4*7 miles. 



TELEPHONE AND TELEGRAPH PROBLEMS 281 



Ex. 2. The current sent into the above cable is 

Co = sin 5000^ + sin 10000^ (1) 

Show that at the end of 30 miles the current is 
c = 0-04303 sin (5000^ -31458) + 0-01169 sin (10000^ -4-449). (2) 
Show on squared paper (1) and (2) for a complete period; that 
is for the time T= 27r/5000. 

Ex. 3. The case of Ex. (2), but the cable having also / = 0-l 
henry. Show that the current in 30 miles becomes 

c' = 0-3933 sin (5000^ - 10-6) + 0-3933 sin (lOOOOif - 21 -2) (3) 

Show (3) on the same sheet of squared paper as (1) and (2). 

Note that (2) is quite unlike (1) in shape, whereas (3) is exactly 
like (1) in shape. This is what we mean by no distoiiion. It is 
curious that distortion as great as what (2) shows should not 
prevent telephonic transmission. 

156. Submarine Telegraphy. Assume the following cables to be 
infinitely long ; the sending voltage is 30 sin 60^ ; what is v and 
what is the current when «=1000 nautical miles in each easel 
r and k are given per nautical mile. In each case take Z = 0, s = : 





r 


k 


n 


'- or l/zo 
r 


Suez- Aden cable 


10-42 


0-3580x10-6 


0-01058 + 0-01058t 


0-001435 [45°] 


Aden -Bom bay - 


7-02 


0-3610 xlO-« 


-00872 + 0-00872t 


0-001753[45°] 


Persian Gulf, 1864 - 


6-25 


0-3486x10-6 


0-008085+0-0080851 


001830[45°] 


Atlantic, 1865 - 


4-27 


0-3535 X 10-6 


0-00673 + 0•00673^• 


0-002229[45°] 


French Atlantic, 1869 


3-16 


0-4295x10-6 


0-00638+0-00638^ 


0-00-2852 [45°] 


Direct U.S. cable - 


2-88 


0-4095x10-6 


0-005948+0-0059481 


0-002920 [45°] 



f=VI(i^^>=V7[*n 



In each of these cases 

in 

li n = a + ai, V = SOe~ "^ sin {qt - ax), c = -v. 

I find the following results for a; =1000 nauts (or 1000 nautical 
miles) : 





' 


c 


Suez-Aden 


0-000764 sin (9« 


-606^) 


0-000001 1 sin (5« - 


-561°) 


Aden-Bombay - 


-00490 sin (5« - 


-500°) 


0-0000086 sin (gr 


-455°) 


Persian Gulf 


-00924 sin (g« - 


-463°) 


0-0000169 sin (g^ - 


-418°) 


Atlantic, 1865 - 


-03583 sin (g« - 


- 386°) 


0-0000799 sin (g^ - 


- 341°) 


French Atlantic 


0-05085 sin (5^ - 


366°) 


0-0001450 sin (g<- 


- 321°) 


Direct U.S. 


07832 sin (g«- 


-341°) 


00002287 sin {qt - 


- 296°) 



282 ELEMENTARY PRACTICAL MATHEMATICS 

It must be evident to students that in telegraph and telephone 
cables, if I and s are so small that we may neglect them, the received 
currents are the same if L\/rk or Lhk is the same, L being the 
length of the cable. In fact if B is the whole resistance of a cable 
and K is its whole capacity, then the received cun-ents are the same 
if EK is the same. But we have usually no such rule in telephone 
lines, because I and 5 are not negligible ; if Hs great enough any 
musical note lags for the same time as another of quite different 
pitch. The lag as an angle is gx, but the lag in time is gx/q. 

I do not like here to interrupt the electrical problems. The 
student will find two heat problems analogous to the above electrical 
ones in Arts. 162 and 163. 

157. Returning now to the more difficult case of a line of limited 
length L miles. As we have two arbitrary time functions A and B 
we must have two conditions given. See Art. 153. 

Ex. 1. Let v = Vq where x = 0, and let i; = where x = L. That is, 
let the line be put to earth at x = L. This is an easy exercise. It 
is to be remembered that when a; = 0, cosh nx = 1 and sinh nx =■- 0. 

(3) gives v^^ = A+0 or A=v^), and also = A cosh nL + B sinh nL. 

Or B= -Vr,^—, — -. Hence we have everywhere 
^smh nL ^ 

, coshTiL . 1 \ /,, 

cosh nx — ^^r — ^smh nxV (1 ) 

c= -y&mhnx — ^-^ — j.cosh?«cj-^ (2) 

T ^-11 1' ^ r + lqi 

1 still use - to mean ^ or Z(.. 

n n ^ 

Ex. 2. Let the given conditions be that both Vq and Cq are known 

n T 

when x = 0. Here A=Vq. Also Cq= --B or B= — Cq, so that 

T n 

T 

v = Vq cosh nx — Cq sinh nx, (1) 

c = — Vq sinh nx + Cq cosh rta; (2) 

Ex. 3. Let it be known that v = Vq where x = 0, and also that at 

V 

x = L, where v is v^ and c is Cj, we have — =B, the resistance of a 



v = v,(^ 



telephone or recorder or other receiving instrument. Note that B 
may have unreal terms. 



TELEPHONE AND TELEGRAPH PROBLEMS 283 

Taking formulae from the last example, 

V 

v^ = Vq cosh nL — Cq sinh nL, 

Cj = -~Vq sinh iiL + Cq cosh nL. 
Dividing and putting the quotient equal to B, we find c^ to be 
cosh iiL + R- sinh nL \+R- tanh nL 



R cosh nL-^- sinh nL 
n 



R + - tanh nL 

n 



The expressions for v and c can now be written out. It will be 
found that the current through the receiving instrument or 

^1= 2^ (1) 



R cosh nL + - sinh nL 



a very useful formula. In practical cases, where L is usually 
large, sinh nL = cosh nL, and then 

(2) 



^ + - I cosh nL 

nj 



Ex. 4. It is to be remembered that 



i? 4- - tanh nL 

^0 i+i^^tanhTiZ 
r 

v V 
In a long line tan rtL is 1, so that whatever R may be, — = ~ or Zq. 

It is, however, worth while to work a few exercises on this. 

If r = 88, A: = 0-05 x 10"^, Z = 0, s = 0, we have already found that 

when 2 = 5000, 71 = 0-1482 [45°] and - = 593 [-45°]. 

Take lines of length L= 1, 5, 10, 20, 30 miles, and take i? = 0, 50, 
100, 200, 400 ohms. In each case calculate vjc^ from (5). 



Values of vq/cq for 


L 


R=0 


R=bO 


«=100 


i£ = 200 


/J = 400 


1 

5 

10 

20 

30 


88-0[-0°-2] 
429-4[ - 10°-3] 
669 -7 [-32° -7] 
602-4[-46°-5] 
591-1 [-45°] 


138-0[-0°-7] 

473-0[-12°-6] 

675-8[-34°-9] 

603-3[-43°-9] 

591-2[-45°] 


188-0[-l°-4] 

513-3[-15°-l] 

677-8[-36°-9] 

604-0[-44°-l] 

591-4[-45°-0] 


287 -7 [-2° -9] 
585 0[-19°-9] 
677-0[-40°-3] 
603-1 [-44° -5] 
591 -6 [-44° -9] 


485-6[-5°-7] 
696-0[--29°-l] 
665-1 [-44° -8] 
6020[-45°-l] 
593 -U- 44° -9] 



284 ELEMENTARY PRACTICAL MATHEMATICS 



Note that when a line is long, whatever R may be, -9 becomes 
nearly -. This is evident from (5), for if L is large, tanh^iL is 

nearly 1, and then ^ = -. 

c, n 

Ex. 5. Standard cable A of Ex. 12, Art. 149. From (2) of 
Ex. 3 calculate the current c-^ received by an instrument whose 
^ = 420 + 0-084ge [that is, its ohmic resistance is 420 with an 
inductance of 0'084 henry], for the following values of q and 
L = 40, 60, and 80 miles. Take v^ = 1. As 

Cj = 1 sin qt ^ (420 4- 0-OMqi + - j cosh Lit, 

the values of coshi?i given in Ex. 10, Art. 149, will be found useful, 
r 30028 



\lq 



(l-^). 



s 


Ci for 40 miles. 


Ci for 60 miles. 


Cj for 80 miles. 


3000 

5000 

7000 

10000 


70-04 xl0-Ssin(9«- 169°) 
35-71 xlO-6sin (5(5 -240°) 
17-21 xl0-6sin(g^- 300°) 
5-893 xlO-^sinig^ -.377°) 


15-09 xl0-6sin(^^- 263°) 
4-385 xlO-«sin(g^- 360°) 
1 -441x10-68111(^/5 -443°) 

0-3024 xl0-6sin(g«- 547°) 


2-976 X 10-6 sin (g«- 355°) 
0-5396 X 10-«sin(^<-480°) 
0-120-2 X 10- 6sin(g<- 585°) 
001563 X 10-6sin{g^-7l7°) 



Ex. 6. Submarine cable ; per nautical mile, r = 3, A' = 0*373 x 10"*', 
/=.0, s = 0, 2 = 34 (this corresponds to 95 letters per minute), 
VQ=\^\nqt. There is a recorder whose resistance is 344 and 
inductance 10*1 at the receiving station which is iy = 2500 miles 
away ; what is the current Cj through the recorder coil 1 Here 

i2 = 344 + 344i, n = 0-006 168 [45°], and - = 486-3 [ - 45°] = 344 - 344i, 

Z7i=15-42[45^]= 10-90+ 10-90i, cosh X?i= 27 1 00 [624°]. 

Hence, c^ = 1 sin qt + 688 cosh Ln. 

If the number of letters per minute is increased to 168, so that 
we take q = 60, find the current c^ through the recorder coil. 



Letters 
per minute. 


1 


ci 


95 
168 


34 
60 


5-363 X 10-8 sin (g^- 624°) 
l-732xl0-»8in(/7<-860°) 



It has been found by practical experiment that the study of the 
sending of a sine current with ?;q = 30, where the arriving Cj is 
160 X 10~^, tells us that when reversed dot elements are sent in 



TELEPHONE AND TELEGRAPH PROBLEMS 285 

actual word signals, the maximum sending voltage being 30, the 
maximum currents in the dots received are about 160 x 10"^ amperes. 

Ex. 7. At the end of cable E of Ex. 9 of Art. 139, with r = 2-88, 
/ = 0, ^ = 0-4095 X 10~^, 5 = 0, /> = 2432 nauts, there is a recorder of 
resistance 263 ohms with inductance 4*4 henries. Between the 
cable and the recorder is a Varley condenser of 40x 10"^ farad. 
If Vq = 30, find Cj the current through the recorder (P*) without the 
condenser, (2""^) with the condenser, when ^ = 60 and when q = 4:. 
Notice that we wish to receive as little of the slowl}^ varying current 
as possible, and this is the object of employing the Varley condenser. 

L (1) liq = 60, with the condenser as ?i = 0-00841 [45°], 
cosh 2432 n = 952820[828°], - = 263 - 263i, 
the resistance of the Varley condenser is 

Ci = 30sin^^^(-417i + 263 + 263i + 263-263i)coshZri. 
This denominator is 
(526 - 4172) coshi.?i = 671[ - 38°] x 952820[828°] = 6-393 x 10«[790°], 
therefore c^ = 4-692 x 10-« sin (qt - 790°). 

(2) Without the condenser 

Cj = 30 sin qt -^ 526 cosh Ln = 5-985 x 10"' sin {qt - 828°). 

II. (1) If ^ = 4, with condenser, the resistance of the condenser is 

62502, 



4x40xl0-« 
7i = 0-002172[45°], cosh Zw = •20-93 [2 14°], 

- = 938 - 938i, i2 = 263 + I8i. 
n 

The denominator is ( - 6250i + L'63 + 1 8i + 938 - 938i) cosh L7l 
or (1201 - 7l70i) cosh Lti^ 1-521 x 10-'[133°], 

so that Ci = 1-972 x lO'^sin (qt - 133°). 

(2) If ^ = 4 and without condenser, the denominator is ' 
(263 + 182 i- 938 -9382) cosh Zti or 31660[176°], 
so that Ci = 9-475 x 10"* sin {qt - 1 76°). 

Thus we see that the Varley condenser reduces the slowly varying 
current to the one-fifth part of itself, whereas it reduces the signals 
only by one-fifth. 

Ex. 8. In the famous Submarine Telegraph Relay, the receiving 
rean'der of resistance r is shunted by a great inductance L, the ohmic 
resistance of which i§ insignificant. A current C from the cable 



286 ELEMENTARY PRACTICAL MATHEMATICS 

divides itself so that c goes through the recorder and the rest 

through the shunt. It is evident that = 1 - t--. I shall only take 
amplitudes or effective currents. ^ 

Thus, let Z = 6 henries, r = 200 ohms. 

When 2 = 60, ^=0-7681. 

When g = 4, ^=0-07974. 

That is, of the good signalling currents the instrument receives 
77 per cent., whereas, of the bad slowly varying currents, it only 
receives 8 per cent., the remainder passing through the shunt. 

158. Exercise. To find the best winding of the receiving instru- 
ment at the end of a fairly long line. Ex. .3, Art. 157, gives the 
current received. But in all such instruments we desire current 
turns to be great. Now when any instrument is wound with 
different sizes of wire, if the insulation is always in the same pro- 
portion to the copper, p = at'^, X = bt^ if p is the ohmic resistance, 
A the inductance, a and b constants, and t the number of turns. 

r 
In any case, let - = a - pi. 

n 

The B of Art. 157 means p + Xqi, so that the denominator is 

{p + Xqi i-a- /3i) cosh nL. 

Neglect the cosh nL, as our winding cannot affect this part ; to 
make the current turns great, we must make the amplitude of 
{p-\-a + i(Xq- 13)} ^t small 



or 



at + J + iibtq - jj small. 



The square of this amplitude is 



(-?)H%-fy 



To make this as small as possible, we evidently ought to have 
%-f = 0; 

that is, Xq = l3. To make at + ~ a, minimum, it will be found that 
at^ = a or p — a. 

(1) What is the best p and A for a telephone receiver at the end of 
a considerable length of standard cable, r = 88, ^ = 0*05 x 10"^, / = 0, 

.9 = 0, 7 = 5000? Ans. We found - = a -/3i = 419-5 - 419-5i 

' ^ n ' 



TELEPHONE AND TELEGRAPH PROBLEMS 287 

419*5 
Hence /» = 419*5 ohms, ^= ^rvrj/x =0"084 henries. 

It is to be remembered, however, that eddy currents in the iron 
of the instrument (which is usually not nearly well enough divided) 
cause the static resistance of such a telephone receiver to be only 
100 or 120 ohms instead of 419-5. 

(2) What are the best p and A for a recorder coil at the end of a 
considerable length of submarine cable ; r = 3, Jc = 0*373 x lO"*', / = 0, 
s = 0, 2 = 34^ 

Here n = s/S x 0*373 x lO'^ x 34^, 

w = 0•006168^/^ = 0•006168[45°], 

Hence the recorder coil ought to have a resistance of 344 ohms 
and an inductance of %*/- or 10*1 henries 

159. When the receiving instrument has a resistance r^ and it is 
in parallel with a resistance r^, then 



B = — ^-^ and c^ = ■ 



A most important part of Brown's submarine relay is the great 
inductance r^, which shunts his recorder r^. 

It is easy to state c^ where there is a condenser in the r^ part, and 
instead of this we may imagine a condenser through which c^ passes 
before the shunt is reached [this is the famous Varley condenser], 
and compare such methods of working. 

160. At the sending end of the line we may not be given Vq. 

Suppose there is a source of alternating current whose e.m.f. is E 

E 
and resistance it^, then Vq = E -BqCq or Vq= v. . So we replace 

Vq in (2) of Ex. 3, Art. 157, by this expression. In long lines r^ may be 

taken to be - or Zq. Thus the current c^ received by an instrument 

E 
of resistance R Sit L miles will be Cj = 

Whatever be the contrivance at the sending end of the line there 
is an alternating electromotive force E in it, and knowing the 



288 ELEMENTARY PRACTICAL MATHEMATICS 

resistances it is easy to express v^^ in terms of E and r^^ , and therefore 
to express the received current in terms of E. 

The above formulae suit at once a submarine cable, because the 
return conductor, the sea, may be taken to have no resistance. 
But in telephony, the return conductor is a separate wire about a 
foot away, of the same resistance as the going conductor. The 
formulae are correct for this case if 

r = the resistance per loop mile. 
/ = the inductance per loop mile. 
s = leakance per loop mile. 
Jc = capacity per loop mile. 

These statements mean : Suppose we have two lengths of the 
conductor AB and CD placed side by side at the same distance 
apart as the real going and coming ; and each is 1 mile in length. 
Then r is the resistance of AB and CD in series. That is, the 
resistance of two miles of conductor. The self-induction of the 
circuit ABDC, if the ends were joined, is I; or Ms the mutual 
induction between AB and CD. s is the leakance between AB and 
CD, and k is the capacity between AB and CD regarded as the two 
coatings of a condenser. 

L is the distance from transmitter to receiver ; the length of the 
conductor is of course 2L. 

I need hardly say, that although I have given as the title to this 
chapter "Telephone Problems," the formulae apply even more 
directly to problems on the transmission of power by alternating 
electric currents. But it is only on long lines that the effects of 
distributed capacity are great. 

161. I recommend electrical students to read the papers published 
by Prof. Kennelly, of Harvard, as he has greatly developed this way 
of making calculations. In particular, his paper published in the 
Proceedings of the American Institution of Electrical Engineers in 1904 
contains some interesting practical examples. He uses terms, how- 
ever, which the student may have difficulty in understanding, nor 
indeed do I understand why he uses such terms. 

My B he calls z^, "the impedance of the receiving apparatus." 
The denominator in equation (1) of Ex. 3, Art. 157, or 

r 
B cosh Ln + - sinh Zn 
n 

he calls ^j, "the receiving end impedance." 



TELEPHONE AND TELEGRAPH PROBLEMS 289 

My - or rather i- he calls Zq, "the sending end impedance." 

My ?Q or -2 he calls Zs, "the impedance of the circuit at the 

sending end." 

The amplitude of my Vq he calls E, " the maximum cyclic E.M.F. 
of the sending apparatus"; that is, he assumes that the sending 
apparatus has no resistance or impedance. I have found in the 
usual arrangement of transmitting apparatus on the common battery 
system, that the errors due to this wrong assumption may be con- 
siderable. See Art. 151. 



P.M. 



CHAPTER XXXVI. 
HEAT PROBLEMS. 

162. The following heat problem is analogous with the electrical 
problem of Art. 146. 

The temperature v at the hot surface of the metal of a steam- 
engine cylinder follows the law 

Vq sin qt ( 1 ) 

Assume the periodic time to be ^ a second (as if the engine made 
2 revolutions per second), so that ^ = 27r x 2 = 6-2832. I shall use 
C.G.s. units. The conductivity of the iron is ^ = 0*20, the capacity 
for heat of one cubic cm. of iron or ws = 0-87 ; find v at any depth x 
and any time t. Assume that the surface of the metal is quite 
plane instead of being cylindric, and that the metal is infinitely 
thick. Here 



'4- 



-^ qi = S-7 + 3-7 i and v = e~"*Vo sin g^, 
or leaving the sin qt out, as being always understood, 

'= -^^ = ^^^""^0 (2) 

Now e"^ = e-3-7-3-7^ = g-s-v* [- _ 3.7^]^ 

Hence v = VQe-^-^'' sin (qt - S -7 x) (3) 

and c = nkv. 

We may, if we please, introduce another term, and instead 
of (3) use v = VQe-^-^''sm{qt-3-7x) + ax, 

because v = ax is also a solution of the fundamental equation. 

It is easy to see how this satisfies the case of a steady flow of 
heat inwards as from a steam jacket. I will, for the rest of this 
exercise, assume that a = 0. 



HEAT PROBLEMS 291 

If the heat enters the surface from an atmosphere at the tempera- 
ture V, make some assumption as to the current of heat which 
would enter under the influence of the difference of temperature 
V-Vq. If e is the emissivity of the surface, the simplest assumption 
to take is e{V-VQ) = GQ. Neglecting the steam jacket term, that is, 
taking a = 0, c^ = nkv^^ so that 

e{F-VQ) = nkvQ or V^v^'"^ % 



■< 



1 w^\ 
1+-K 



If there is a wet layer on the metal in a steam cylinder, e is 

very much larger than when the surface is dry, as it is when highly 

k 
superheated steam is used. Take two cases: (P*) - = 6, a dry skin; 

k ^ 

(2''^) - = 0-1, a wet skin. 

P*case. F=(l + 6n)vQ = {23'2 + 22'2i)vQ = 32'U[i3°'74:]vQ. 
or r= 32-1 lV(,sm{qt + 43°-74). 

That is, the range of temperature in the steam is 32-11 times 
the range of temperature in the skin of the metal. 

2'''*case. V = {V37 + 0-37i)v^=l'i2l[lb°-l]vQ. 
or r=l-421?;oSin(2^-l-15°-l). 

That is, the range of temperature in the steam is only r421 times 
the range of temperature in the skin of the metal. 

We may desire to know the amount of heat which enters the 
metal in one cycle. If at the surface c = CQsinqt-b, the negative 
current b being due to the steam jacket, the amount of heat entering 
in the positive part of one cycle is Cq/2^ - ^bT, where T is the periodic 
time, in this case 0-5 second. 

This enables an approximation to be made to the amount of water 
missing as indicated water, because of condensation. [See my book 
on Steam, pages 381-9.] 

163. The following heat example is mathematically the same as 
the last, and is also the same as that of the telephone and telegraph 
cables of Arts. 154-6. 

Lord Kelvin and Principal Forbes buried thermometers at various 
depths in the rock at Craigleith Quarry, Edinburgh. The changes 
of temperature were (P*) of 24 hours period, (2'"*) of one year period. 
In such work as the above we ought therefore to study two terms 



292 ELEMENTARY PRACTICAL MATHEMATICS 

with two values of 5-, the temperature v at any depth being the sum 
of what each term produces. But in this case it was possible in the 
observations to get results for the yearly period only, and these 
were the results : 



Depth in feet below 
the surface. 


Yearly rauge of temperature, 
Fahrenheit, 


Time of highest 
temperature. 


3 feet 

6 feet 

12 feet 

24 feet 


16138 

12-296 

8-432 

3-672 


August 14 

„ 26 

Sept. 17 

Nov. 7 



[Observations at 24 feet below the surface at Calton Hill, 
Edinburgh, showed the highest temperature on January 6*^, but I 
am not now studying the Calton Hill observations.] 

I am sorry that as I write I cannot refer to the original paper by 
Kelvin and Forbes, and I do not recollect how they reduced their 
observations. A more tedious and more accurate method than the 
folloAving might be adopted. First, multiplying the yearly range by 
5 and dividing by 18, I get the amplitude in the Centigrade scale. 
Convert feet into centimetres. Write the lag in days, first assuming 
that the lag at the three feet depth is m days. Assume our theory 



to be true, and then 



v^e'"^ sin (qt - ax). 



X 

feet. 


X 

centimetres. 


Centigrade. 


lag in 
days = d. 


ax the lag in 

radians 
calculated. 


d as calculated 

if theory is 

correct. 








^'0 











3 


91-44 


4-483 


m 


0-207 


12 + 


6 


182-88 


3-4156 


m+12 


0-414 


12+12 


12 


365-76 


2-3422 


w + 34 


0-828 


12 + 36 


24 


731-52 


1-020 


m + 85 


1-656 


12 + 84 



Plotting logio,-?/ and x (feet) on squared paper, I find that the 
points lie very fairly well on a straight line [this verifies part of our 
theory], and I take logio?/ = 0-72 + 0-03a; or y = 5'25e-°'^% x being in 
feet. Hence a = 0*069. The lag in radians therefore ought to have 
the values calculated above. If the lags d given in days are correct, 
d _ lag in radians 
365-25 "" ^27r ' 

and we can therefore calculate what d ought to be if our theory is 



HEAT PROBLEMS 293 

correct. I find the answers given above. It is seen that the 
numbers agree with those observed, very nearly. The maximum 
temperature ought to have been found at the depth of 12 feet, not 
on Sept. 17 but on Sept. 19. And Nov. 7 ought to be Nov. 6. 
These are not great discrepancies. 

If X is taken in centimetres, cue = 0*207 when ic = 91-44, so that a 
in proper (or C.G.s.) units is 0*002264. Now we know that 






^^ = 2-264 X 10-3. 

The periodic time is 1 year or T= 31-56 x lO*' seconds; q is 27r/T 
or 2= 1991 X 10"'; ws the capacity for heat, of the rock, per cubic 
cm. may be taken as 0*5. We have then 

2-2642X 10-« X 2^ = 0-5 x 1-991 x 10"^ 

This gives X- = 0*00951, the probable conductivity of the rock at 
Craigleith Quarry. 

I find that the result published by Kelvin and Forbes is 0-01068. 
They may have taken a different value of ws from mine ; or they 
may have carried out the above work more carefully than I have 
done. 

Students of electric cables are fond of talking of wave-lengths. 
Here the heat wave-length is 27r-^a or 6-2832-^0-069 or 91 feet. 
That is, at a depth of 91 feet the lag is just one year ; would such a 
student say that heat is therefore found to travel in rock with the 
speed of 91 feet per annum ^ It seems rather absurd, and yet this 
is the very thing that he says about the speed of electricity 
in cables. 

It is quite easy to consider heat problems which are analogous 
with that of the electric cable of limited length. 



CHAPTER XXXVII. 



VECTORS. 



164. Hitherto I have only considered scalar quantities, quantities 
which are dealt with like mere numbers. 

A vector quantity, like dis- 
B placement, velocity, momentum, 
acceleration, force, impulse, strain, 
stress, flux or fluid, flux of electric 
current, magnetic force, magnetic 
induction, etc., has direction and 
magnitude. A vector can be re- 
presented by a straight line ; its 
magnitude to some scale by the 
length of the line ; its clinure or 
mi by the clinure of the line; its sense by an arrow-head. Thus 
the length of the line AB (Fig. 50) represents a vector to some scale 
of measurement ; its sense is shown by the arrow-head. 

If A'B' is drawn parallel to AB and of the same length and sense, 
it represents exactly the same vector. 





Fig. 51. 



165. Addition of Vectors. If a, b, and c (Fig. 51) represent 
three vectors, to add them : make them the sides of a polygon 



VECTORS 



295 



(Fig. 52) ; take care that their arrow-heads are circuital [follow my 
neighbour]. Thus AB is the same as a, CD is b, EF is c. 

Then the last side of the polygon with a non-circuital arrow-head 
represents their sum. Or, as we write it, 

AB + CD + EF=AF or & + \) + c = AF. (1) 

The student ought to take three or four 
vectors and add them according to the above 
rule, taking them in quite different order as 
sides of a polygon ; in every case the same 
answer is obtained. In fact a + b = b + a. We 
apply now the same symbols as we use in 
algebra, that is in dealing with scalar quan- 
tities, and say that (1) is the same as 

Si + \) + c-AF=0 (2) 

or a. + \} = AF-c (3) 

Notice that when a letter is used for a 
vector we employ Clarendon type. Notice that 
our vectors are not necessarily all in one plane, 
but when they have all sorts of clinures the 
polygon is a gauche polygon, and must be illustrated by bits of wire. 

We may say that all the above statements make up our definition 
of a vector. 

Vector quantities are such quantities as may be added and 
subtracted according to the above rule. 




Fig. 52. 



Fig, 53. 



It is a self-evident truth that a displacement is a vector. If we 
say that a point is moved to the east 1 foot and to the north-east 
2 feet, we draw AB and BC to represent these two displacements, 
and we see (Fig. 53) that the vector sum of the two, AC, gives 
the real sum of the two displacements. 



296 ELEMENTARY PRACTICAL MATHEMATICS 



Here AB + BC^AC 

or AB = AC-BC 

or BC=AC-AB, 

so that subtraction of vectors is as easy to understand as addition 
of vectors. 

It is evident that the sum of the displacements AB, BC^ CD, DE, 
and BF (Fig. 54), is exactly the same as that of AG, GH, and HF ; 
and AF expresses it. 




Pig. 54. 



Similarly a velocity, being displacement in unit time, is a vector. 
We look upon the statement that all the above quantities are vectors 
as a self-evident truth. "^^ There is nothing else to be done. We 
can no more prove the truth of such a statement than the truth of 
our own existence. As soon as we comp-ehend the statement we see 
that it requires no proof. In the same way, acceleration, being 
velocity added per second, is also a vector. Force is mass multiplied 
upon acceleration, and is therefore a vector also. In so far as a 
force has an actual position as well as mere clinure, it has a property 
in addition to that of a mere vector. 

Notice that when we use a letter a to represent a vector, it is in 
Clarendon type, and when we see Clarendon type in an equation 
we know that it is a vector equation without special telling. Also 
- a is just the same as a, except that its sense, its arrow-headedness, 
is reversed. 

When I was young I spent much time over Duchaylas' proof that 
forces are added in the above way. The old 15 page trouble of 
our youth has disappeared from the books, I am happy to say. It 

•^ This statement is too sweeping. Advanced students will find that things 
not vectors may be mistaken for vectors. Thus o. finite rotation is not a vector. 



VECTORS 297 

was not only illogical, but stupid, and everybody now recognises 
this fact. When will the remaining so-called "proofs by abstract 
reasoning " of the school-books disappear 1 Unfortunately they have 
a specious appearance of being useful in mind-training, and the 
stupid teachers can teach nothing else, and so Euclid and its trailing 
cloud of miseries can disappear only gradually.* 

166. It is very difficult to set questions on this subject, because 
it is so difficult to state clinure or direction. If I use the points of 
the compass, only a few students will understand. I am afraid that 
I must ask students to remember the following way. Let OX 
(Fig. 55) be a given standard direction; say, towards the east 
from 0. Now if I always assume the sense of a vector OF to be out 
from 0, and if I state the angle XOP as traced out aw/i-clockwise, 
will the student be able to understand ? 




Fig. 55. 

Thus, as drawn in Fig. 55, I take XOP to be 50° ; XOF, 120° ; 
XOP", 180° ; XOF", 230° ; XOP"'\ 320°. That is, I never measure 
angles in the clockwise direction.! 

*I look upon the study of Euclid as one of the most valuable of post- 
graduate studies. What I object to is that every English boy should be forced 
to pretend that he can follow Euclid's reasoning. 

fin passing, I may say that in specifying a system of forces not passing 
through a point I use the following method : Let O be a known point and OX 
the standard direction. Let the direction of a force AB cut OX in the point 
A and let AB he the sense or arrowheadness ; let the angle BAX be called d 
and the distance OA be called a; if P be the magnitude of the force, then the 
force is specified by aP^. Thus, for a force of 50 lb., angle ^=125° ; OA =2, 
we use the symbol 256i25-. 

Hitherto, to specify forces in setting questions, examiners have directed 
squares and triangles to be drawn, specifying their forces by means of these 
figures ; the result has been that students believe problems in statics to have 
an occult connection with regular geometrical figures. 



298 ELEMENTARY PRACTICAL MATHEMATICS 

It will now be easy to set questions. I shall state the magnitude 
of each vector ; assume that its sense is out from 0, and I shall state 
the angle it makes with OX. 

It A = 20j4oo ; B = 1 0300 ; C = 30280" ^ D = 1 2330= ', E = 20470. 

The answers must be stated in the same way. Find by con- 
struction : 

(1) A + B + C + D + E. A71S. 30-55..5. 

(2) A + B + C-D-E. Ans. 32-6220O-8- 

(3) -A-B-C + D + E. Ans. 32-6400.8. 

(4) -A-B-C-D-E. Ans. ^0-b^^..,. 

As the letter A represents a vector, there is much meaning 
wrapped up in one letter; clinure and sense and the scalar part, 
which is mere magnitude. If we say Q = 1 JP, it can only be true 
if Q and P are of the same clinure (or mi some people call it) 
and sense. 

It is very well worth while to express a vector P as PqP, where 
Pq is the scaler magnitude, or tensor as it is sometimes called, and p 
is said to be the unit vector. 



167. If a vector a changes to b in the time U, we see that the 
vector AB (Fig. 56) has been added in this time, and we may say 

that there has been an average increase 
at the rate AB -f U per second during 
this time. 

We here say a -}- AB = b 
,or a + Sa = b, 

and — - is the average rate of increase. 

This is a vector in the direction AB. 

But if we want to know the actual rate 
of increase at any instant, we must take 
an interval of time U which is smaller 
and smaller without limit. 

The student might consider a very 
simple case ; a body moves in a circular 
path of radius r at a constant speed v. Consider the body as it 
proceeds from a point a in the time U to the position h. Let 
the distance ah be called 8s. Draw lines (Fig. 57) OA parallel to 
the tangent at a, and OB parallel to the tangent at h and make 




Fig. 56. 



VECTORS 299 

them equal in length, each representing the velocity v to some 
scale. 

Then, according to the above rule, the vector AB-^U is the 
acceleration. 

The anejle ^ 0J5 = — = -^- and AB = vx-^^, 

so that acceleration = — in a direction at right angles to v at any 

instant, that is towards the centre of the circular path. 

It is quite easy to extend this to show that the total acceleration 
of a point moving with changing speed in a curved path, is the 
vector sum of the centripetal acceleration and of the 
acceleration of speed. 

The study of tortuosity is made comparatively easy by 
vector algebra. 

168. Multiplication of Vector Quantities. If I were 
asked to multiply 2 tables by 3 chairs, I would not refuse ; 
I would say 6 chair-tables. But if I were asked to say 
what I mean by a chair-table, I would refuse to answer; 
because nobody has ever given a meaning to the term. 
But I do know that when this sort of thing comes into a 
Physical Problem we can always give a useful meaning. 
This is beyond ordinary Algebra, and yet our processes 
are carried on by the rules of Algebra. Observe that I do 
not saj^ that O 

2 tables x 3 chairs = 3 chairs x 2 tables. ^°* ' 

Whether this is or is not true depends upon the meaning we attach 
to the whole process. It is easy to multiply any quantity whatever 
by a mere number. For example, 2 tables x 3 = 6 tables. Also a 
velocity of 6 feet per second due east multiplied by 3 is 18 feet per 
second due east. Again, if we multiply a vector by any scalar 
quantity we get a vector in the same direction, although the meaning, 
the name of its unit, may not yet have been fixed. A velocity of 
6 centimetres per second due east multiplied by 3 grammes is a 
momentum of 18 gramme-centimetres per second due east. Thus, 
then, the multiplication of a vector by any scalar quantity is not 
difficult to understand ; the result is a vector in the same direction. 
But what meaning are we to attach to the multiplication of vectors 
by one another? 



300 ELEMENTARY PRACTICAL MATHEMATICS 

Surely you are all acquainted with examples. If a vertical force 
of 4 lb. acts through a vertical distance of 3 feet, we say that it does 
the work 3 ft. x 4 lb. = 12 foot-pounds. 

Now these are two vectors that I have multiplied together. In fact, 
the product of a force into a displacement is quite familiar to us. 
If there is any force A, and if the body that it acts upon has a 
displacement B, we say that the work done is AB. 

Some people write this S.AB, because work done, energy, is a 
scalar quantity which has no direction and is not a vector, but I 
prefer to leave the S out. Now notice that we use AB to represent 
the work done, whatever may be the clinures and senses of A 
and B. 

It is well worth while to study this example carefully, for this 
product of two vectors exactly illustrates what we mean by the 
scalar product of any two vectors. 

Note that my unit of force is that used by the engineer, the 
weight of one pound at London. 

Let us suppose that the force A is A lb., but it is evidently 
something more; I must state the direction. Let it be vertically 
upwards. If a force of 1 lb. acting vertically upwards be com- 
pletely indicated by the letter a, then 

A = Aa.. 
That is, a is the unit force vector and ^ is a mere numeric. 

Again, suppose B (Fig. 58) to be a displace- 
ment of B feet in some direction which may 
not be vertically upwards ; suppose that the 
letter b represents the vector displacement of 
1 foot in that actual direction of B, that is, 
that the letter b signifies direction as well as 
1 foot, in fact g =, ^ 

Then we can say that 

AB = ^^.ab. 

The product of two mere numbers A and B 

is easy enough, but what signification are we 

to give to the product ab of two unit vectors ; 

a being one pound vertically upwards and b a displacement of 1 foot 

in some direction not necessarily vertically upwards ? 

It is exactly our chair- table question. We are asked to give a 




VECTORS 301 

meaning to the product of our units ab. Now we know that 
meaning if in the above case we are going to say 

AB = work done. 
Because the work done when the force A acts on a body when the 
body has the displacement B is obtained by finding the resolved 
part of A in the direction of B (we call this A cos Q lb. if Q is the 
angle between the vectors), and multiplying by B feet, our answer 
being AB cos Q foot pounds. Or we might have done it by finding 
the resolved part of B in the direction of A (this is B cos B feet), 
and multiplying by A pounds, our answer as before being ABcos 
foot pounds. 

Since then aB = ^^ . ab = AB cos foot pounds, 
we see that the product of 

a, which is a force of 1 lb. vertically upwards, 

b, which is a displacement of 1 foot making an angle 6 with 

the upward direction, 
is cos 6 foot pounds. The student should here note that ab is the 
same as ba. 

Leaving out of account mere names of unit quantities, we may 
say then that our definition of the meaning of the scalar product of 
two unit vectors is ^^^ ^ 

where is the angle between their directions, and the meaning of 
the product AB is AB cos 6, where A and B are the tensors and 6 is 
the angle between the directions of the vectors. To measure 6, 
always draw the two vectors from a point with their arrowheads 
going out from 0. As we deal only with a cosine of this angle, 
it will be found that it may be measured either clockwise or 
anti-clockwise. 

Ex. 1. A force of 350 lb. acts on a tram-car in a direction 
towards 20° west of north; the velocity of the car is 20 feet per 
second due north (or ^ = 20°). Find the Power being given to 
the car. (Generally when we multiply any kind of force by any 
kind of velocity we are in the habit of calling the answer "the 
activity.") 

Ans. 6578 foot pounds per second. In this case the activity is 
merely what the mechanical engineer calls power. 

Ex. 2. In the above case the force acts towards due west 
(or 6 = 90°). What is the power or activity 1 Ans. 0. 



302 ELEMENTARY PRACTICAL MATHEMATICS 

Ex. 3. In the above case the force acts towards the south-west 
(or (9= 135°). Find the power. 

Ans. Minus 4949 foot pounds per second. 

Ex. 4. In the above case the force acts towards the south 
(or d = 180°). Find the power. 

Ans. Minus 7000 foot pounds per second. 

169. We see that if OX (horizontal) and OY vertical are two 
standard lines at right angles in a plane, the position of a point F 
may be defined in two ways. 

(1) If we know x its distance to the right of OF and y its distance 
above OX. 

(2) If we know r its distance from the point and the angle 6 
which r makes with OX. 

We see that the connection between these co-ordinates is 
x = r cos 6, y = r sin 0, and of course r^ = ic^ -i- y"^. Also yjx = tan 6. 
Given x and y we can find r and dy or given r and 6 we can find 
X and y. 

170. When we deal with vectors in all sorts of directions in space 
it is necessary "^ to use three standard directions, OX, OY, and 
OZ mutually at right angles A vector OP is projected as OA in 
the direction OX. If a, ^, y, are the angles which OP makes 
with OX, OY, OZ, then OA = x=OP .cosa, OB = y = OP .cos p, 
OC=z=OP . cos y. Two of these three angles ought to be given if 
we want to specify the direction of a vector. 

Now it is easy to show that OP^ = OA^ + OB^ -{-OC^ so that 
OP^.cos^a+OP^cos^P-hOP^cos^y = OP^ or cos2a-f-cos2^-f cos2y = 1. 

171. Hence, when a man, in telling me the direction of a vector, 
gives me its a, /3, and y, I have one check on the accuracy of his 
measurement. 

Ex. 1. I have carefully measured and found in a certain case 
that a = 20°,/? =75°, 7 = 76°-5. 

Calculate what y ought to be from the other two and state the 
percentage error in the measurement of y, if the others are quite 

correct. Ans. 77°*08, percentage error 100 x ' ^^ or 0*75 per cent. 

77'Uo 

*0f course I do not mean that it is necessary. A single vector a, when 
given, enables us to specify all vectors in space parallel to a. Given another 
independent vector b (or parallel to b), we can specify any vector of any size 
in the plane of a and b (or in any parallel plane). Thirdly, any third 
independent vector c enables us to specify any vector whatever as xa + yTa + zc. 



VECTORS 



303 



Ex. 2. The following values of OP and of a and /3 are given for 
three vectors. Find y in every case. Find the projections in the 
three standard directions. Add each set up. 



Values of 
OP. 


a 


/3 


vas 
calculated. 


OA 


OB 


oc 


30 
25 
15 


70° 
150° 

85° 


37° 

84° 
170° 


60° -3 
60°-7 
8r-4 


10-26 
-21-65 
1-308 


23-96 
2-612 
- 14-77 


14-86 
12-23 
2-242 


Sums 








- 10-08 


11-80 


29-33 



Ex. 3. Find the vector whose three projections are the above 
sums. 

Alls. OP = v/(10-08)2 + (11-8)2 + (29-33)2 = 33-18, 

and it makes angles a, ^8, and y with OX^ OY, and OZ such that 
cos a = - 0-3038 or a = 107°-7, 
■ cos^= 0-3556 or ^= 69°-2, 
cosy= 0-8840 or y= 27''-9. 
We have therefore found the sum of the above three vectors. 
The student who knows descriptive Geometry will find the sum of 



z 


C 






y^ i 


P/ 




Q 


7or^\^j^^^ 


^N 








B Y 



Fig. 59. 



the above three vectors by construction, and see if he gets the same 
answer. 

The projection of a vector OF upon the plane XOY (Fig. 59) is 
the line OQ or OF cos FOQ. The projection of OF upon the line OZ 



304 ELEMENTARY PRACTICAL MATHEMATICS 

is the line OC or OP cos POZ. Let POZ be called 6 (I called this y 
just now) ; we have seen that POQ is the complement of 6, 

0C= OP. cose, 0Q= OP. sine. 
Let the angle QOX be called <f) ; then OA = 0Q. cos <^ and 
0B==0Q. sin <^. If we let the length of OP be supposed to be 
always positive and we call it r; if OA is called x, OB, y; OC, z; 
these being the projections of r on OX, OY, and OZ, the three 
mutually rectangular standard directions, we have two ways of 
stating our results, both of which are in use. 

a^ = rcosa or a; = rsin ^. cos^; 

y = rcosP or y =^r sine, sin <j); 

z = rcosy or z = rcose. 
It will be noticed that x^ + y^ + z'^ = r^ 
and cos^ a + cos'^/3 + cos^ y = L 

172. If we think of the plane XOY as the equatorial plane of the 
earth, and OZ as the axis of the earth, then the position of a point P 
in or on or outside the earth, which moves with the earth, is given 
if we know its r, or distance to the centre of the earth, its e or 
co-latitude, and its <^ or east longitude. <f> is evidently the angle 
which the plane containing OZ and OP and OQ (called a meridianal 
plane) makes with the standard meridianal plane ZOX (in our 
analogy this is the meridian through Greenwich). It is to be 
remembered, however, that the a, /3, y way is also very greatly 
used to specify direction, cos a, cos ft and cosy are called the 
" direction cosines " of a line, and the letters I for cos a, m for cos ft 
n for cosy are very greatly used. So that x = lr, y = mr, z = nr, and 

We usually specify the inclination of a plane by stating the 
direction cosines of its normal, and of the surface of a body at any 
place by the direction cosines of the normal to the surface at that 
place. 

I see that I have instinctively left the consideration of vectors in 
general, and applied my rules to mere displacement vectors. In 
fact, we have at length reached the subject of Geometry; that is, 
the relation of mere lines to one another. If we consider XOZ, 
ZOY, and YOX as three mutually perpendicular standard planes of 
reference, the position of a point P is fixed if we know its x, y, and z; 
or its r, e, and </> ; or its r, and any two direction cosines. And if 



VECTORS 305 

we know the position of a point in any one of these three ways, we 
can calculate the other dimensions. 

173. Ex. 1. The X and y co-ordinates of a point in a plane are 
3 and 4 ; find r and 6. As r cos ^ = 3 and r sin ^ = 4, 

t = ^^^^ = tan^ or 6> = 53°-l 3 nearly ; 7-2 = ^2 + ^2 = 25, 
3 ?• cos 6^ "^ ' ^ ' 

so that r = 5. 

Ex. 2. If a point has f=10 inches, ^ = 37°, find x and y, 
a; = 10 cos 37° = 7-986 inches, i/ = 10 sin 37° = 6-018 inches. 
Now going to three dimensions and Fig. 59 : 
Ex. 3. If r = 10 inches, d = 35°, <^ = 62°, find x, y, z. 
Here «= 10 sin 35'cos 62° = 2-693 inches, 

y^\0 sin 35° sin 62° = 5-064 inches, 

^ = 10 cos 35° = 8-192 inches. 
Ex. 4. If « = 3, ?/ = 5, ^ = 6, find i\ 6, and <^. 
Here r- = x'^ + y'^-\- z^, so that r = 8-370. 

= rcos(9 or cos 6' = — - = 0-7171 or 6^ = 44°-18; 

3 = 8-370sin44°-18xcos<jE) 
or cos </) = 3 -^ (8-370 x 0-6969) = 0*5144 

or </> = 59°-03. 

Ex. 5. If £c = 3, 2/ = 5, ;? = 6, find r, /, m, n. 

Answer as in last case, r = 8-370, 1 = x/r or 0*3585, m = y/r or 0*5975, 
7i = 3/r or 0-7171. 

174. Exercises on Scalar Products. (1) The flow of fluid per 
unit area through a surface is nV, where n is the unit vector 
normal to the surface, and V is the velocity of the fluid. If the 
fluid flows at 5 feet per second at an angle making 35° with the 
normal to the surface, find the flow per square foot. 

Ans. 5 cos 35° or 4-096 cubic feet per second. 

(2) If E is electric force and D is electric displacement, it is 
important to calculate half of ED. If E = 250 vertically down- 
wards, and if D = l-56 making an angle of 42° with the downward 
direction, find JED. Ans. J x 250 x 1 -56 x cos 42° = 1 +4-9. 

(3) If A and B are the sides of a parallelogram, A -f B and A - B 
are the diagonals. 

(A + B)2 = A2 + -2AB + B2. Show that this is really the ordinary 
formula for the square of the length of a diagonal. 

(A - B)2 = A2 - 2 AB + B2. Show that this is really the ordinary 
formula for the square of the len'gth of the other diagonal. 
P.M. U 



306 ELEMENTARY PRACTICAL MATHEMATICS 

Express the geometrical meaning of 

(A + B)2 + (A-B)2 = 2(A2 + B2) 
and (A + B)2-(A-B)2 = 4AB. 

(4) If A, B, and are the edges of a parallelepiped, show that 
the ordinary formula for the length of the diagonal is given by 

(A + B+C)2 = A2 + B2 + C2 + 2AB + 2AC + 2BC. 

(5) If a plane figure moves, it traces out a volume (per unit of 
its area) equal to the scalar product nD, where n is the unit vector 
normal to the plane area and D is the displacement of the centre of 
area. If the area keeps parallel to itself (in the final position it may 
rotate about an axis through the centre of area, and this will make 
no change in the answer), n is the same in every position, and D 
will then be the resultant or total displacement. 

175. When I was asked what I meant by 

2 tables x 3 chairs = 6 chair-tables 
I refused to answer. Algebra had done its duty; it could go no 
farther. I may now give a meaning to 

1 chair-table 
and invent a new science. But I may give another meaning and 
invent another science, and both sciences may be useful. So when 
we multiply vectors we can give a scalar meaning to our answers, as 

we have already seen. But there 
is no reason why we should not 
give another meaning, a vector 
meaning, and that in certain cases 
this also should be of value. 

There is another way* in which 
the product of vectors enters into 
calculations. Suppose we have a 
magnetic field B. That is, it is 
of the amount B in the direction 
OA (Fig. 60), and amount and 
direction are both specified by the letter B. Let there be a 
conductor in the field with a current C flowing in it. Here again 
C signifies that there is a current of the amount C flowing in the 
direction OG. We know that there is a mechanical force acting on 

*Any combination in products of the components of two vectors might 
perhaps be called some special kind of product of the vectors ; but our two are 
the only useful ones ; furthermore, as they are independent of the axes of 
co-ordinates, they are the ones that occur in Nature. 




VECTORS 307 

the conductor in the direction OD, which is at right angles to both 
OA and OB, and its amount is ^(7 sin d. If this force is denoted by 
F, we say that p is the vector product CB 

or F=r.CB. 

Definition. The vector product of two vectors and B is a vector 
at right angles to both, its tensor being the product of the tensors 
of C and B multiplied by the sine of the angle between them ; that 
is, the area of the parallelogram of which OA and OG are two sides. 

Exercise. A body is spinning about an axis OX at A radians 
per second, and the letter A represents both amount of spin and 
direction OX. In fact A is the tensor of A, and A has the direction 
OX. A point P in the body is at the distance r = OF from 0, and 
the letter R signifies the amount r or OF and also the direction of 
OF. It is evident that the velocity V of P at any instant is specified 
both in magnitude and direction by 

V=r.AR, 

for the amount of it (its tensor) is Ai' . sin XOF, and its direction is 
at right angles to the plane XOF. 

It is outside the scope of this elementary book to dwell further 
upon the two kinds of products of vectors which enter so much 
into physical calculations. I may not even dwell upon the rule 
for the sign of F. AB, or show that 

r.AB=-r.BA. 

As in other mathematical methods, the student has to get accus- 
tomed to the use of a few formulae. For example, what is aFbc^ 
We have the vector product of b and c, and we have the scalar 
product of this with a. If a, b, and c are the three edges of a 
parallelopiped meeting at a corner, aFbc is evidently the volume 
of the parallelopiped, and hence aFbc = bFca = cFab, a set of very 
important identities. Again the student ought to show that 
VcFaib = a . be - b . ca. 

I do not know of any book introducing the student to the use of 
Vector Algebra. An exceedingly interesting (but much too short) 
introduction will be found in Mr. Oliver Heaviside's Electromagnetic 
Theory, Vol. I., and electrical students in particular will find it 
valuable, for it is by means of Vector Algebra that electromagnetic 
theory can be most easily studied. 



BOAKD OF EDUCATION EXAMINATIONS. 

Previous to 1912 the examinations were in Stages 1, 2, and 3. In future 
there will be no examination in Stage 1 . There will be what is called 
the Lower examination ; it is of the same standard as the old Stage 2. 
There will be what is called the Higher examination ; it is of a standard 
higher than the old Stage 3. Examination papers for the years 1910, 
1911, and 1912 are here given, with the answers, and where they are 
necessary, with suggestions as to methods of working. 1 do not give the 
papers of earlier years, as almost every question in them has been 
incorporated in this book already as an example or exercise. 



1912. Lower Examination. 

1. The four parts («), (6), (<?), and {d) must all be answered to get full 
marks : 

(a) Without using logarithms compute by contracted methods 
3-207 X 0-01342 -^9-415. 

(6) Using logarithms compute the square root of 

62-41 xO-1352-r 2-416. 
(c) State the values of the sine, cosine, and tangent of 230°. 
{d) State the value of the Napierian logarithm of 13520. 
Am. (a) 0-004571 ; (6) 1-869 ; (c) -0-7660, -0-6428, 1-1918 ; (d) 9-5118. 

2. The three parts (a), (b), and (c), must all be answered to get full 
marks : 

(a) A hollow circular cylinder of cast iron is 10 inches long and 

3 inches inside diameter ; what is the outside diameter if the 
cylinder weighs 30 lb. ? [One cubic inch of cast iron weighs 
0-26 lb.] 

(b) ABC is a right-angled triangle, C being the right angle. If ^ C is 

4 inches and the angle A is 40°, find BC and the area of the 
triangle. 

(c) If ^ is 1 -201, find i(e* + e-*). 

Ans. (a) As Tf^=30 = 0-7854(Z)2-^2) ^ 10 x 0-26, we have 
30 
^'~^^ 2-6xQ-7854 ^^^'^^> ^' = 23-69, i)= 4-867 inches. 



J^tK';j^\ 



BOARD OF EDUCATION EXAMINATIONS 309 

(6) As ^ = tan 40 =0-8391, 5(7=3-3564. 

The area is 3-3564 x 2 = 6'7128 sq. inches, 
(c) 1-813. 

3. The three parts (a), (6), and (c) must all be answered to get full 
marks : 

(a) The sum of .v and i/ is 5*17 and the sum of their squares is 14-25 ; 
find X and y. 

(6) What is the area of the curved surface of a right cone if its base is 
3 inches in diameter and vertical height 5 inches ? 

(c) There are two perfectly similar statues of marble ; the height of 
one is 2-13 times the height of the other : the smaller weighs 20 lbs. 
What is the weight of the other ? 
Ans. (a) As a;+i/=5'l7 and a:^ + i/'^ = l4'25, we have 

2^3/ = 5-172 -14-25 = 12-48. Kence a;^-2xi/-\-f = l-77. 
That is, a;— y=l'33. Adding this to x+t/, we get 

2^=6-50 or ^=3-25 and y=l-92. 

(6) The slant height is \/25 + 2-25 = 5-22. The circumference of the 
base, which is the length of arc of a sector in the developed curved 
area, is 9*4248 inches ; the area is ^(9-4248 x 5-22) or 24-6 sq. inches. 

(c) Volun]Les_ofsimilar things are as the cubes of their like dimensions, 
andf" heigE^ are proportional to volumes if the materials are the 
same: — Thcrlarger weighs 2*13^ times or 9-664 times the other ; that 
is, 193-28 lb. 



4. If i/=x^-3'39x-{-l'96 for values of x from to 3, plot sufficient 
points of the curve on squared paper to show for what values of x, y is 0. 
What are these values of ^ ? Ans. 2-655 and 0-735. 

5. When the pointer of a planimeter is guided once round the 
boundary line of a plane figure, the reading of the instrument R is such 
that the area A is CR^ where C is some constant. 

If R is 22-48 for a circle of 3 inch radius, what is (7, the area being 
required in square inches ? On applying the instrument now to an 
indicator diagram, R is found to be 3-77 ; what is the area ? The length 
of the diagram being 4-11 inches, what is its average breadth ? 

Ans. 1-258 ; 4*742 sq. inches ; 1*1538 inches. 

6. A disc whose outside radius is r^ and inside radius r^ is rotating ; 
the radial stress P and the hoop stress §, at any radius r, are 

$=ro2+V+'-^^-0-538r2. 

If 7*0=10, '/•i = 4, write out the expressions for P and Q. Now calculate 
the values of P and Q, for the following values of r : 4, 6, 8, 10, and show 
them in two curves. 

Am. P=ll6-l^-r2, ^ = 116 + ~^-0-538r2. 



310 ELEMENTARY PRACTICAL MATHEMATICS 



r 


P 


« 


4 





207-4 


6 


35-56 


141 1 


8 


27 


106-6 


10 





78-2 



When these are plotted I am afraid that more points are needed to 
show the curve P properly. It is evident that P is a maximum when 
r = 6-325. 

7. The insulation resistance R of a piece of submarine cable is being 
measured ; it has been charged and the voltage v is diminishing according 
to the law v = be- 'Z^'^, 

where b is some constant, t is the time in seconds ; K is known to be 
0-8 X 10-6. 

If V is noted to be 30 and in 15 seconds afterwards it is noted to 
be 26-43, find It 

Ans. When ^=0 let v = 30 ; when t = \b let v = 26-43, 
^ = 306-'/-^^ so that 26-43 = 306-15/^'-. 

Therefore ^^^^3 = ^' 

30 
S=15-=-A'log.^g:j3; 

working this out, we have i2=148 x 10^. 

8. X and y are as tabulated. It is known that 

find u approximately in the middle of each interval. Show y and u as 
two curves, x being abscissa : 



X 





01 


0-2 


0-3 


0-4 


0-5 


0-6 


0-7 


0-8 


0-9 


1 


y 


5-000 


5-736 6-428 


7-071 


7-660 


8-192 


8-660 


9-063 


9-397 


9-659 


9-848 



and y need not be plotted to the same scale. 

Ans. Placing x and y in columns, I find in each interval : 



X 


y 


Fx 


V, 


0-05 


5-368 


7-36 


100-44 


0-15 


6-082 


6-92 


99-61 


0-25 


6-750 


6-43 


98 05 


0-35 


7-366 


5-89 


95-73 


0-45 


7-926 


5-32 


92-83 


0-55 


8-426 


4-68 


88-93 


0-65 


8-862 


4 03 


84-61 


0-75 


9 230 


3-34 


79-55 


0-85 


9-528 


2-62 


73-84 


0-96 


9-754 


1-89 


67-67 



BOARD OF EDUCATION EXAMINATIONS 311 

The values of y here tabulated for the intermediate times are the means 
of those given in the question. Thus 5'368 is ^(5-000 + 5-736). 

9. A pin at one end of a horizontal lever is 42 inches from the fulcrum. 
If the lever turns upwards 70°, find 

(i) the length of the path traversed by the pin, 
(ii) its distance from its original position, 
(iii) its height above its original level. 

What would these answers have been had the angle turned through 
been only 7° ? Ans. 51-31, 48-17, 39*47 ; 5-132, 5-124, 5-120. 

10. There is a probability that if a man stands at so short a distance as 
d from the muzzle of a gun which discharges a projectile of weight w^ 
his sense of hearing will be hurt. If d is proportional to the sixth root 
of w and if o? is 10 feet for the discharge of a 64 lb. shot, what is d for 
the discharge of a 9 lb. shot ? 

Ans. d=cw^j c being a constant. Therefore 

c=10-r 64^ = 5 and rf=5x9^ 

so that the answer is 7-211 feet. 
Perhaps this explains why so many artillery men are deaf. They 
think that because their ear is not hurt at, say, 11 feet from the muzzle 
of a large gun, they may come quite close to a small gun with impunity. 

11. A square whose side is 4 inches has one diagonal parallel to an 
axis which lies in the plane of the square at the distance of 3" from the 
diagonal. The square revolves about the axis and generates a ring. 
What are the volume and area of the ring ? 

Ans. 301-6 cubic inches ; 301-6 square inches. 

12. The ends of a round barrel are 40 inches in diameter and the mid 
section is 48 inches in diameter : the barrel is 60 inches long. What is 
its volume ? For what shape of barrel is your approximate rule quite 
exact? 

Ans. The average section is J(40^+402 + 4x 482) + 6, and 60 times this 

is the volume, or 97,513 cubic inches. If x is distance along the 
axis of the barrel from any point in the axis, for Simpson's rule 
to be correct we must have the cross-section A following a law 

A=a + bx-{- cx^y 

where <x, h and c may have any values. Of surfaces of revolution 
there are several for which the rule is exact, but the only one (I 
believe) that suits the barrel shape is the ellipsoid of revolution. 

13. A ship going at 21 knots changes its direction steadily from due 
North to North-west in two minutes : what is the radius of its path ? 

The angle 45° is 0-7854 radians = arc -i- radius and arc = 21 x 2-f-60, so 
that r=21 X 62jj-^ 0-7854 =0-891 nautical mile or 5420 feet. 

14. The lengths of the intercepts of a plane on the three mutually 
perpendicular axes are 0^ = 3, OB =4, OC=b. Find the length OP oi the 
perpendicular from the origin on the plane. Ans. 0/^ = 2-164. 



312 ELEMENTARY PRACTICAL MATHEMATICS 



1912. Higher Examination. 

1. The four parts (a), (6), (c), and (d) must all be answered to get 
full marks : 

(a) Without using logarithms, compute by contracted methods, so that 
four significant figures shall be correct — 

10-32 X 0-005231 -^ 0-02076. 
(6) Using logarithms, compute cosh a and sinh a, where 

«= 1-013. 
(c) Using logarithms, compute 

(3-062 -^27-15)-l•23. 

{d) If i means >/- 1, express - 2-35 + 1-962 in the form r(co8 6 + isi.n 6) 

and extract its cube root. 
Ans. (a) 2-600. 

(6) As e« = 2-754 and 6"" = 0-3632, cosh a = l-5586, sinh a = 1-1954. 
(c) 14-65. 
{d) See Art. 146. 3-060(cos 140°-] 67 + ^ sin 140°-167). 

One of its cube roots is 

l-452(cos46°-72 + isin46°-72) or 0-9956 + 1 0561. 

2. When the pointer of a planimeter is guided once round the 
boundary of a plane figure, the reading R of the instrument gives the 
enclosed area in units which depend on the position S in which the roller 
frame is clamped on a graduated arm. For a circle of 3 inches radius 
and for three positions S, the readings are as follows : 



s 


24 


27 


30 


R 


25-29 


22-48 


20-23 



There is a simple relation connecting >S' and R^ find it. Find also the 
two positions S for which the readings will be the area in square inches 
and in square centimetres respectively. 

Alls. It is well known that SR is constant ; but a candidate ignorant 
of this would probably try plotting log S and log R ; and he will 
find that SR=607. The area of the circle is 28-274 square metres 
or 182-42 sq. cm. ; dividing these into 607, we find the two positions 
S to be 21-47 and 3-33. 

3. The following values of a; and y were observed in a laboratory, and 
theory suggested that there might be a law 

y = aoo-\-h log X. 
There are errors of observation. Using squared paper, try if there is 
such a law and, if so, find the most probable values of a and 6. You may 
take log X as being log^) x. 



X 


10-2 


31-0 


52 


75 


104 


132 


181 


y 


3-76 


6-26 


7-99 


9-54 


11-39 


12-94 


15-67 



BOARD OF EDUCATION, EXAMINATIONS 313 

Ans. In any equation connecting x and y, if there are only two constants 
it is possible to plot things which will give a straight line. Thus, 
in this case we can 

(1) Plot - and ° and get a straight line. 

(2) Plot ^ and i^i^ and get a straight line. 

X X 

(3) Plot -^— and r-^ and get a straight line. 
I find y = 0-045^+ 3-3 logio^. 

4. If y = ^e**, what is -^ '^ An electric condenser, of capacity K farads 

(xX 

and leakage resistance R ohms, has been charged, and the voltage v is 
diminishing according to the law 

dv _ V 
di~~~KR' 
Express v in terms of the time, t seconds. If A''=0'8 micro-farad 
(that is, 0'8 X 10~^ farad) ; if y is noted to be 30, and 15 seconds after- 
wards it is noted to be 26 '43, find R. 

See Art. 113. If y = J e«% ^ = aAe"'' = ay. 

Hence v^v^fi'*^^^ or ^oge — = ^^- 

If ^=0 when i; = 30, evidently ^0 = 30; therefore 
, 30 15 „ 15 

logH5;7^ = T^ or R= 



26-43 KR ^j^g 30 



26-43 



The Napierian logarithm of -i^ is 0-1266, 

15 X 10^ 
^ = 7^ — c^.A^aa = 1 48 X 1 0^ ohms = 1 48 megohms. 

5. A periodic function is given either as a curve or in a table of 12 or 
18 or 24 or more equidistant values ; describe how you would find the 
values of the various terms of its Fourier Series. 

One answer to this is given in Art. 132. See also page 323. 

6. There is a value of x between 2 and 2*5 which satisfies the equation 

l-5j7- -^^ + 3 sin - - 5 logio^=2'6. 
Find it, correct to three significant figures. Ans. 2-340. 

7. Solve ^+2/|+...=o. 

Take 7i2 = 200,/=7-485 ; let ^=0 and also ^=10 when ^=0. 

See Arts. 121 and 133. 

Ans. The auxiliary equation is m2+2/w + ^^=0, and its roots are 

in this case m = - 7*485 ± 1 2i if i=\I -\. 



314 ELEMENTARY PRACTICAL MATHEMATICS 

Therefore x = e-'^*(^ sin \'2.t+B cos 120- 

If ^=0 when ^=0, it is evident that 5=0, and if so, ^ 

^= Je-7-485«( _ 7-485 sin 12^+ J 2 cos 120- 

If -1- = 10 when «f is 0, ^ = t^ = 0-8333. 
at ' 12 

Hence x=0-SZZZe-'^-^*^uil2t. 

8. The rate per unit increase in volume at which a pound of gas is 
receiving heat during its expansion is 

dH I ( dp \ 

-dv=^iVdv^yn 

If pv'^ = c. find -^ in terms of p. y, n, and c are constants. 

dv ' 

JTT 

For what value of n is -r- = ? 
dv 

Ans. p = cv~'^; therefore ,-= -?icv"""~^= —??.-, 

,, , dp , o?iy y-7i 

so that v-f-=—np and -j-= ' — ^p. 

dv ^ dv y-l^ 

This is when it = y, and the expansion is then said to be adiabatic. 

9. Describe a method of finding whether a given curve or the tabulated 
values of x and i/ follow, approximately, the law 

i/=a + bjf^ 
or i/=b{x+ay 

or y = a + he'^. 

This question is answered in Art. 79. 

10. In a telephone line of length I, where g- is 27r/, if /is frequency or 
pitch of a musical note ; let r, the resistance per mile, be 88 ; let ^, the 
capacity per mile, be 0"05 x 10~^. Take 9^ = 5000. Let n = ^rkqi, where 
i means v -1. Let ^ = 40 miles. If ^=100 + 0045^1 be the resistance of 
the receiving telephone, the current through it is 



(7=2Fo-(/2 + ^)^% 



where V^ is 10 sin 5000^. C is of the form a sin {qt + 6), where q is 5000. 
Find a and h. 

[Suggestion: a + /3i may be put in the form p(co8 + t8m6); if this 
operates upon m sin qt the result is mp sin {qt+6). Note also that 

e^' = cos f3 + i sin B.] 
See Art. 146 and Chap. XXXV. 

w=\/88 X 005 X 10-6 X 50001 = 0-1483 v7=0-1049 + 0-1049i = 0-1483[45°], 

^=M4S45-]=^93[-45-] = 419-419,-. 



BOARD OF EDUCATION EXAMINATIONS 315 

Hence i^ + - = 100+200^■ + 419-419^ = 519-219^ = 563[-22°•9], 

^w = 4•196 + 4•196^ and e'" = e«96g4i96i 
Now 4-196 radians = 240°'43 and e'*'^^=66-42, so that 6^'' = 66-42 [240° '43]. 

11. We wish the current-turns in a certain telegraph coil to be as 
great as possible, and this will be the case when the following expression 
is a minimum : /^ SOoy /Lq 500\2 

The resistance of the coil is i2 = afi ohms, where a is a constant and t 
is the number of turns of wire. The inductance of the coil is L = ht^. 
The constant q is known. 

Show that we get the best result when ^=500 and Lq — bQO. 

This is proved in Art. 158. It is, however, well for the practical man 
to remember that ^=500 ohms may mean a resistance for steady constant 
currents of 120 ohms, there being a spurious resistance due to eddy 
currents and hysteresis in the iron or steel. 

12. A telephone transmitter has a varying resistance 

/2= 10 + 0-1 sin 5000jf, 
the variation being due to a musical note ; what is the frequency or 
pitch of this note? If there is a battery of electromotive force E 
which is 6 volts, the current is E-^R. Show that the current has a 
varying term of the above frequency, but that there are an octave and 
higher harmonics which are quite insignificant because 10, the constant 
part of R^ is large. This explains why the common battery system gives 
better articulation than the local battery system. 

Am. The current is 6+(10 + 0-l sin 50000=0-6+(l +0-01 sin 5fX)00. 

Now, as a sine can never exceed 1 or be less than — 1, calling 
0-01 sin 5000^ = a, we may say that 

0-6-^(1 + a) =0-6(1 - a) = 0-6 - 0-006 sin 5000^ 

That is, the varying part of the current is what we desire it to be. 
The answer is really 0-6(1 -a + a^ — a^ + etc), and the a^, a^ terms really 
mean the introduction of terms like sin 10000^, sin 15000^, etc., that is, 
the octave and higher harmonics ; but it is seen that they are utterly 
insignificant in this case because a is so small, and this is due to 10, the 
constant part of R, being large. 

13. The lengths of the intercepts of a plane on the axes are 0J=3, 
OB— A, 0C=5. Find the length of OP, the perpendicular from the 
origin on the plane. Find the angles which OP makes with the axes. 

Ans. 0P=2-n, a = 43°-7, /? = 57°-25, 7 = 64° '35. 

14. The curve ^ = l+0'2x^ rotates round the axis of x, generating a 
surface of revolution. What is its volume between the cross-section at 
^=0 and the cross-section at ^= 10 ? 

Ans. The integral of ttj/^ or 7r(l + 0-4.1-2 -|-0-04^'^) jg 



/ , 0-4 „ 0-04 .\ 
taking a;=10, the answer is 943-37r or 2963. 



316 ELEMENTARY PRACTICAL MATHEMATICS 



1911. Stage 2. 

1. The four parts (a), (6), (c), and (d) must all be answered to get full 
marks : 

(a) Without using logarithms, compute by contracted methods, correct 

to four significant figures, 

23-56 X 0-1023 -^ 2-363. 
Compute to Jive significant figures, and reject the last. A^is. 1*020. 

(b) Using logarithms, compute the cube root of 

1782^0-3152. Ans. 17-81. 

(c) State the values of sin 26°, cos 110°, sin 220°. 

Ans. 0-4384, -0*3420, -0-6428. 

(d) A man is 68 inches in height ; what angle does he subtend at the 

distance of 300 yards ? Why are the sine and tangent of this angle 
and the angle itself, stated in radians, practically the same ? 

[If a figure of a very small angle is drawn, the reason can be put 
in a few words. The distance is 300x36 or 10,800 inches. The 
angle in radians is 68 -M 0800, and we multiply by 57*3 to get it in 
degrees. Ans. 0*3608 degree or 21 65 minutes.] 

2. The three parts (a), (6), and (c) must all be answered to get full 
marks : 

(a) A hollow sphere of outside diameter D and inside diameter d. 

What is its volume ? What is its weight W if the material weighs 

w lb. per unit volume ? 

The weight Wis 10 lb., the inside diameter is 6 inches. What is 

the outside diameter if w is 0-3 lb. per cubic inch ? 
(6) What are the factors of ^2 -3.25^ + 1*56? 
(c) The area of a rectangle is 22-4 square inches and its perimeter is 

19*2 inches ; what are the lengths of the sides ? 
A71S. (a) The volume of a sphere is ^ttt^ or 0'6236d^. Hence 
W=0'523Gw(jD^-d^). 

Putting IF=10, d=6, w=0-3, we have 10=0*1571 (i)^- 216) or 

2)3-216 = 63*65 or /)3 = 279*65 or />> = 6-539 inches. 
(6) The factors cannot readily be seen by inspection. Let the expression 

be (.v — a){x-b); then, if it is put equal to 0, the roots of the 

equation ^2 — 3-25.^ + 1-56 = are a and 6. Now the roots of this 

quadratic are 2-664 and 0*586. Therefore the factors are a;- 2*664 

and .r- 0*586. * 
(c) Let x and 3/ be the lengths of the sides. We have .ry = 22-4 and 

a; + 7/ = 9'6. Squaring x+y, 

^2+2^+^2 = 92*16, 
4ry =89*6. 
Therefore x^ - 2xi/ + 7/^ = 2*56, 

'.r-y=l*6, 
^+y = 9*6, 
2a? = 11 '2 or a? =5*6 inches, 
2y = 8 or y = 4 inches. 



BOARD OF EDUCATION EXAMINATIONS 317 

3. If sin a^-r-x is 0-75, where x is in radians, what is xl Use squared 
paper. 

Ans. The equation is sin ^ — 0-75.^ = 0. Let y = sin ^ — 0-75^. Take 
values of x and calculate y. Looking at your table, which gives 
X and sin x, you see that x cannot be small ; if you are quick in 
seeing whether ^x is nearly the same as sin.t-, you will probably 
begin at about 65 degrees, and you are naturally led to work 70° 
and 75°. 



Angle in 
degi-ees. 


' 


sin a: 


y 


65° 

70° 
75° 


11345 
1-2217 
1 -3090 


666 


0-0556 

0234 

- 0-0159 


73° 


1-2741 


0-9563 


0-0007 



Plotting the three values of ^ and x roughly on squared paper, you 
will be directed to try ^ = 73, and you calculate ^ = 0-0007. As the curve 
between .r = 73 and ^=70 is nearly straight, you may take it to be 
really straight, as you are so nearly right ; or by arithmetic ; — for a 
difference of 3 degrees, the difference in y is 0-0227 ; for how many degrees 
is % = 00007 ? A ns. 3 x ofy or 009 degree. The answer is then 73° - 0-09 
or 73°-09. I might have worked in radians instead of degrees. The 
answer is 73°-09-^57-3 or 1-276 radians. 

4. Part of a roof has an area of 150 sq. feet ; its inclination to the 
horizontal is 37° : what is the area of its plan ? Prove the rule which 
you use. Atis. 150 x cos 37° or 119-8 sq. feet. The proof is well known. 

5. The following corresponding values of x and y were measured. 
There may be errors of observation. Test if there is a probable law 

and, if this is the case, what are the probable values of a and h 1 



X 1-00 


1-50 


2-00 


2 30 


2-50 2-70 


2-80 


y 


0-77 1 1-05 

i 


1-50 


1-77 


2-03 


2-25 2-42 



Ans. Tabulate the values of x\ plot a^ and ;//, and we see at once that 
the points lie nearly in a straight line. Taking the line that lies 
most evenly among the points, we find 
y= 0-54 + 0-24.^2. 



6. A railway train is at the distance s miles from the terminus at the 
time t hours from starting. Do not plot s and t. What is its average 
speed in miles per hour in each tabulated interval of time ? Assume that 
this is really the speed in the middle of the interval, and uow plot time 
and speed on squared paper. What is the speed and what is the 
acceleration when ^ = 1-07 ? 



318 ELEMENTARY PRACTICAL MATHEMATICS 



103 



105 



1-07 



20-15 20-76 21-42 



1-09 



1-11 



113 



22-13 22-88 23 67 



Alls. Speed, 34*25 miles per hour ; acceleration, 125 miles per hour per 
hour. 

7. If s=10^2^ where s is the space in feet which has been passed 
through by a body in t seconds, find s when ^ = 3. Now find the space 
when t = ^ + m. What is the space passed in the interval m seconds after 
t = Z1 What is the average speed during this interval ? What is this as 
m gets smaller and smaller ? 

5 = 10x9 = 90, 
s + Ss = 10(9 + 6m+m2), 



Ss = 60m+10wi^ 
It 



: 60 + 10m. 



This becomes ;7t = 60 feet per second, the true speed when ^ = 3, more 

and more exactly as m gets smaller and smaller. 

8. The speed of the rim of a centrifugal pump or fan being V feet 
per second, the head H feet produced when Q cubic feet of fluid per 
second are passing seems from theory to be such that 






T' 



The following observations were made on a Bateau Fan : 



Q 


V 


H 


1243 


96-5 


206-5 


825 


112 


371 


46 


96 


144 



Find the values of a, 5, and c. Ans. a = 0-0137, 5=0*0048, c = 0-000272. 

9. Three planes of reference mutually perpendicular meet at 0. The 
distances of a point F from the three planes are .r = 0-25, ?/ = 4-6, 2 = 1*2. 
The distances of a point Q are .r = 0-57, y = 6-82, z = 2'5. What is the 
distance from P to ^ ? A ns. 2-592. 

10. A circle whose diameter is 2 inches rotates about a line in the same 
plane which is 3 inches from the centre of the circle ; thus a ring is 
generated. What is the area of the surface, and what is the volume of 
the ring? Ans. 118-5 sq. inches, 59-23 cubic inches. 

11. The wheel on a vertical shaft is slightly out of balance ; it rotates 
at n revolutions per minute ; the bending moment M has the value 



where tt = 10~^, 6 = 10' 



M- 



l-bn^' 



BOARD OF EDUCATION EXAMINATIONS 319 

What is the critical speed at which the shaft will fracture ; that is, at 
which J/ is exceedingly great ? Ans. w = 316. 

At a speed of twice this critical speed, what is the bending moment ? 

Ans. M= - 133. 
At a speed of half this critical speed, what is the bending moment 1 

Ans. i/=33. 

12. In any class of water turbines, if R is the mean radius of the wheel 
where the water enters, if H is the fall in feet and if P is the total power 
of the fall, it is found that „ p^ j^| 

In one case, where 7-*= 100 and H=\0, R was 1'5 feet. Find R for a 
turbine of the same class when P is 250 and H is 50. 

Ans. 1-5 = c X 100* x 10"*, so that c = 0-845. Therefore R = 0%4S>P^ir^, 
so that in the special case i? = 0*7094 feet. 



1911. Stage 3. 

1. The four parts (a), (6), (c), and (d) must all be answered to get full 
marks : 

(a) Without using logarithms, compute by contracted methods, so that 

four significant figures shall he correct, 

23-56 X 0-01023 ^ 0'02563. 

{h) Using logarithms, compute 

(178-2 ^0-31 52)-"'*. 

(c) State the values of cos 110°, sin 220°, tan 220°, sin 320°, tan 320°, 
sin 580°. 

{d) If i means V — 1, express 5 4-4i in the form 
r(cos ^ + isin ^). 
Now extract its square root. 
Ans. (a) Working to five figures and rejecting the last, I get 9*404. 

(b) The logarithm of 178-2 -f 0-3152 is 2*7523. This multiplied by 

-0-4 is -1*1009 or 2-8991, so that the answer is 0-07927. 

(c) -0-3420, -0*6428,0*8391, -0*6428, -0*8391, -0*6428. 

(rf) As rcos^=5, rsin(9 = 4, tan(9 = 0*8, (9 = 38°*66, and 7-2 = 25 + 16 
or r = 6*403. The answer is 6 '403 (cos 38° -66 + ^■ sin 38° '66), and is 
often written 6-403 [38° -66]. The square root is 

r^l cos 2 4-1 sin -I, 

by Demoivre's theorem, or 

2-531[19°-33] or 2-531 (cos 19°-33 + z sin 19° -33) = 2-388 + 0-8377i. 

2. A crank 1 foot in length rotates uniformly, making one revolution 
per second ; the connecting rod is 5 feet long ; state the distance of the 
cross-head from the end of the stroke as a function of the time. Show 



320 ELEMENTARY PRACTICAL MATHEMATICS 

that the motion is very nearly a simple harmonic motion combined with 
one of half the period. 

Ans. In Fig. 61, let OP the crank be of length r, the connecting rod PQ 
of length I, let AQ=s the distance of the cross-head from the end 



Fig. 61. 

of its stroke. Let the angle QOP be 6. Let the angle OQP be cf>. 
In any such problem, if we project a closed figure upon any two 
straight lines, we get two equations. 

Projecting horizontally and then vertically, we have as ^0=^-fr, 

s + lcos(f) + r COB d=l + r,\ . . 



T 

sin <^ = 7 sin 6^, cos <^ - 



lsin(f) — 7' sin 6. 
If we eliminate <^, we get s in terms of r. As 

so that the top equation becomes 

s=^|l_yi.-^sin2(9|+r(l-cos6>). 

Now I is five times r, and we may approximately consider the 
square root term as Vl - a = 1 - |a. We find 

s = — , sin^ ^+r — r cos 6. 
This is the same as 

s=r+ r-, — rcos ^ — -7^ cos 2^. 

Let 9 = qt, where q is the angular velocity of the crank, so that 
we count time from the inner dead-point position, and we have 

s=r-\- .y — r cos qi — ji cos 2qt 

or 5=1- cos qt + 0'05 (1 — cos 2qt), where q = 2ir. 

3. A fly-wheel is rotating at a radians per second at the time 
t seconds. If M is the moment acting, if fa is a fluid friction and is the 
only resistance, and / the moment of inertia of the wheel, 

Take /= 200 and /= 5000. Find J/ if a = 20 + 0*1 sin I2t. 

Ans. ^=1-2 cos 12^. 
dt 

M=^ 4000 + 20 sin 1 2t -\- 6000 cos 1 2U 



BOARD OF EDUCATION EXAMINATIONS 321 

This may be put in the shape (I take \/e66^+W to be 6000) 
i/=40004-6000sin(12« + 89°-81). 

4. Find an expression for the area of the curve 

1/ = a + bx + cx"^ + ke"*^ + h sin qt. 
Show clearly why you integrate i/. What is the meaning of the 
constant which you add ? 

r c i; --^K 

A ns. A = I y .da;=C+ax+ ^hx^ -\ ^ ^'*+^ + — e"^ — cos at. 

J^ ^ 71+1 m q ^ 

5. Find two values of x to satisfy tan .a? +^=3. 

Ans. ^=1-324 or 75°'875. Also a'=4-64 or 265"'-85. 

6. If R is the resistance, L the self-induction, K the capacity of a 
condenser in a part of an electric circuit, V the voltage between its ends, 
and C the current, show that 



V=(R + Ld + 



IM 



where 6 means -^ if Ms time. If C=asmqt, show that the effect of 
dt 

self-induction in a circuit may be destroyed by putting in a suitable 

condenser. 

Ans. See Art. 127. In this case B=qi, where i is \/-l. As -w^ 

is -=^, it is -^, so that the operator on C is li + iiZq - ^ )• The 

condenser needed is such that Lq — ~^^=0, and K=y-^ is its 
capacity. 



%-"' """ ^^-z^2 



7. The X, y, and z co-ordinates of a point P are 5, 4, and 3 ; find the r, 
6, and cf) co-ordinates. Find also the distance OP and the direction- 
cosines of OP, if is the origin. 

Ans. OP=\/25-|- 16 + 9 = 7071. The direction-cosines are 

5 4 

cosa or ?==;^r=Y =0*7071, cos^ or m=;r-r— -=0*5656, 

3 

cosy or 7i=^-— = 0*4242. 



Also as 5 = r sin . cos </>, 4 = r sin ^ . sin <^, 3 = r cos 0, and we have 

3 
already found r=(?P=7*071, cos ey=;:.-r— =0*4 
4 / U/l 

- = 0*8 = tan</>, so that </) = 38'*66. 



^ =0*4242 and ^ = 64° -9 



8. Suppose A to be a vector quantity and t the time, how do we 
measure dA/dt ? The value of the vector may be measured as «^ where 
a is the amount and is the angle measured anti-clockwise from a fixed 
direction. The vector keeps in a plane. A point has the following 
velocities in feet per second at the following times (seconds) : 



Velocity 


100:«,' 


103*335° 


105*74^ 


107 •25r 


107*862" 


Time - 


10 


10*01 


10*02 


10*03 


10 04 



P.M. 



322 ELEMENTARY PRACTICAL MATHEMATICS 

Find approximately the value of the acceleration when t is 10 02. 

A graphical method is best. 

Ans. Refer to Art. 167. I get 1480i26° feet per second per second. 

9. In a hollow cylinder of nickel steel subjected to internal pressure jt), 
and no pressure outside, when the material is all yielding, if p is the 
radial compressive stress and / the hoop tensile stress at a point whose 
distance from the axis is r, and if f-\-ap — h where a and h are constants 
for a particular kind of steel, and if we also have the usual relation 

find p as a function of r. If the inside radius i\ is 3 inches and the 
inside p^ is 30 tons per sq. inch, what is ^q, the outer radius ? 

[Take for nickel steel a = |, 6 = 30.] 

Ans. Substituting for /from the first equation in the second, 

r^ + (l-«)p + 6 = 0. 

Multiplying by — - and dividing by (1 -a)» + 6, we have 
r 



{\—a)p + h t 
Integrating, log {(1 - a)p + h\ + log r = constant, 

log {(1 - a)p + 6} + (1 - a) log r = constant, 
{(1 — a)jo + 6}r^~" = constant. 

Taking a = f, 6=30, {lp + 30)r^=a 

Now p = 30 when r=3; .-. 37-5 x 3^ = C. 

Where p=p^ = at the outside, 30ro^ = 37-5 x3% 

'37-5\4 



ro_/37;5 
3 V 30 



[The examiner has not asked this interesting question : Find p and / 
everywhere and plot on squared paper. We find that 

p = 15o(?y -120, 

/= 120- 112-5 f?y. 

Such a question is of even greater interest when, as in the case of 
wrought iron and mild steel a = 1. Then p =pi + h log -^ and /= h -jt?.] 

10. A hollow cylinder of cast iron of length 10 inches weighs 62 lb. 
The inside diameter is 0-78 of the outside diameter. If a cubic inch of 
cast iron weighs 026 lb., find the inside and outside diameters. 

Ans. 17=0-26 X 0-7854 x 10(Z)2-c/2) ^2-042 (Z)2_o-6084Z)2), 

62 = 0-79967)2 or 7) = 8-805, rf= 6*868 inches. 



BOARD OF EDUCATION EXAMINATIONS 323 



11, When air or steam is flowing from a vessel where the pressure is pi 

through a rounded orifice into an outside atmosphere, if p is the pressure 

in the orifice, the weight of fluid flowing per second is proportional to 

ar" — x^'^% 

when X is pjp^ and n is -a, known constant, being 0-8850 for nearly dry 

steam. For what value of x is the flow of nearly dry steam a maximum ? 

dW 
Ans. Calling the weight of fluid per second Tf, 



2nx^' 



■^-{\+n)x'\ 



dx 



If we take the case of 



Putting this equal to 0, we have x'^~^ = - 
steam when n = 0*885, we find 

.^-"•^^•' = 1-885/1-770 or ^=0-578. 
That is, there is a maximum flow of steam when the throat pressure is 
0*578 of the pressure inside the vessel, 

12. The value of a periodic function of t is here given for twelve equi- 
distant values of t covering the whole period. Express it in a Fourier 
Series. Terms of the fourth and higher orders are negligible : 2-340, 
3-012, 3-685, 4-149, 3-685, 2-203, 0-825, 0-513, 0-875, 1-085, 1-189, 1-637. 
Ans. I here give the full working out of this question in the mannei- 
described in Art. 132, but towards the end I modify the method, 
because there are few ordinates given. 





1 


X 


A 


B 


c 


D 


E 


F 


G 


H 


«• 


x' 

or 

x-1-1. 


super- 
imposed 
on itself. 


A+B 


Half of 
C being 
sum of 
com- 
ponents 
2 and 4. 


Being I) 
super- 
imposed 
on itself. 


Being 
D+E. 


or com- 
ponent 4 
being 0. 


D-G 
being 
com- 
ponent 2. 



30 

60 



1 
2 

3 

4 
5 

6 

7 

8 

9 

10 

11 


2-340 
3-012 
3-685 

4149 
3-685 
2-203 

0-8-25 
0-513 

0-875 
1-085 
1-189 
1-637 


0-240 
0-912 
1-585 

2-049 
1-585 
0103 

-1-275 
-1-587 

-1-225 
-1-015 
-0-911 
-0-463 


-1-275 
-1-587 
-1-225 

-1-015 
-0-911 
-0-463 


-1-035 

^0-675 

0-360 

1034 

0-674 

-0-360 


-0-5175 
-0-3375 

0-1800 

0-5170 
0-3370 

-0-180 


0-5170 
0-3370 
-0-180 












-0-5175 

-0-3375 

0-1800 


90 
120 
150 


continued 
downwards 







0-5170 
0-3370 
-0-180 


180 
210 


M continued upwards 


-0-5175 
-0-3375 

0-1800 

0-5170 

-0-6630 

-0-1800 


-0-7575 
-1-2495 

-1-4050 
-1-5320 
-0-2480 
-0-2830 


-0-5575 
-1-2495 


240 

270 
300 
330 


1-585 

0-103 

- 1 -275 

-1-587 


0-240 
0-912 
1-585 
2049 


0-600 

0-00 

-0-600 

00 


0-200 

0-00 

-0-200 

0-00 


-1-6050 
-1-5320 
-0-0480 
-0-2830 


Mean \n.^ 
Ordinate/ 




Being 

A 
super- 
imposed. 


Being 

super- 
imposed. 


A + I+J 
being 
3 times 
com- 
ponent 3. 


being 
com- 
ponent 3. 


D 
repeated. 


A-D 
being 
com- 
ponents 
1 and 3. 


N-L 
being 
com- 
ponent 1. 








A 


/ 


J 


K 


L 


M 


N 





324 ELEMENTARY PRACTICAL MATHEMATICS 



To get the A and e of each component in this case, as we have so few 
ordinates, I thought it better to plot the ordinates. Thus for component 2 
we have only three positive ordinates, 0"1800 at 60°, 0'5170 at 90°, and 
0-3370 at 120°. I plotted these and found that A^ is probably 0-527, 
e = 260. And so for the others. My answer is 

^=2-10 +1-633 sin ((^ + 20°) + 0-527 sin (2(^ + 260) + 0-2 sin (3(^+90°). 

13. If r is the radius of the earth and I the distance from the centre of 
the earth to another heavenly body M whose mass is m, then ml{l + rf' 
and m/{l — r)^ are the accelerations towards M at points of the earth 
farthest from and nearest to J/. The tide-producing effects at these 
points are their diflferences from mjl^, which is the acceleration at the 
earth's centre ; prove that the tide-producing effect of M is inversely 
proportional to the cube of its distance from the centre of the earth : I is 
supposed to be very great in comparison with r. 

This proof is given as an example at the end of Chapter V. 

14. The total cost of a certain ship per hour (including interest, depreci- 
ation, wages, coal, etc.) is in pounds 

where s is the speed in knots. Express the total cost of a passage of 

3400 nautical miles in terms of s. What value of s will make this total 

cost a minimum ? Show that at speeds 10 per cent, greater or less than 

this the total cost is not very much greater than what it is for the best 

speed. 

This is like Ex. 5 of Art. 104. The passage lasts 3400/s hours. The 

27914 

total cost is c multiplied by 3400/s or +2-8336--. This is a minimum 

s 

when s = 17 knots. Calculating it for the values of 15-3, 17, and 18-7, 1 find 



s 


Cost. 


15-3 
17 

18-7 


2487 
•2462 . 
2484 



15. In the following table C denotes the radio-activity of a substance, 
t hours after the observations were commenced. There is reason for 
believing that 



dC 

dt 



=aC, 



where a is a constant. 
Try if this is so, and, if so, find the most probable value of a. 



t 


7-9 

i 


11-8 


23-4 


29-2 


32-6 


49-2 


62-1 


71-4 


G 


100 


64 


47-4 


19-6 


13-8 


10-3 


3-7 


1-86 


0-86 



Ans. This is the compound interest law, and if -— — aC^ then by 
4rt. 113 we see that ^^ 

C=CQe<^ or log, (7= log, Co + «^. 



BOARD OF EDUCATION EXAMINATIONS 325 

Using common logarithms, I plot t and log C on squared paper 
and find that the straight line which lies most evenly among the 
points gives log^^ (7= 2 - 0-02933^ 

Multiplying by 2 3026 all across and altering, I find 



1910. Stage 2. 

1 . The four parts (a), (6), (c), and {d) must all be answered to get full 
marks : 

(a) Without using logarithms, compute by contracted methods to four 
significant figures 

5-306 X 0-07632 ^73-15. 
(6) Using logarithms, compute 

(22-15-^4-139)0 8«. 
(c) The value of g^ the acceleration (in centimetres per second per 
second) due to gravity in latitude I is (approximately) 
980-62 - 2-6 cos 2/. 
Calculate this for the latitude 52°. 
{d) The gunners' rule is that one halfpenny (the diameter of a 
halfpenny is one inch) subtends an angle of one minute at the 
distance of 100 yards. What is the percentage error in this rule ? 
Ans. (a) 0-005536. (b) 4-232. 

(c) cos 104°= -0-2419 and g is 980-62 + 2-6 x 0-2419 or 981-25. 

(d) 100 yards is 3600 inches, so that the angle in radians is 
1 +3600. 

To get this in minutes multiply by 57-3 x 60. The answer is 
0-955 minute ; the error is 0-045/0-955 or 4-72 per cent. 

2. The four parts (a), (b), (c), and (d) must all be answered to get full 
marks : 

(a) A hollow cylinder of outside diameter I) and radial thickness t is of 

length I. What is its volume ? If i> is 4 inches and ^ = 0*5 inch, if 
the volume is 20 cubic inches, find I. 

(b) Two similar ships A and B are loaded similarly. B is twice the 

length of J. The wetted area of A is 12,000 square feet and its 
displacement 1500 tons. State the wetted area and displacement 
of ^. 

(c) The cross-section of a stream divided by the wetted perimeter of the 

channel in which it flows is called its Hydraulic Mean Depth. 
What are the Hydraulic Mean Depths when water flows in a pipe 
of diameter d (i) when the water fills the pipe, (ii) when it only 
half fills the pipe ? 

(d) What is the number of which 0*6314 is the Napierian logarithm ? 

Ans. (a) If d is the inner diameter, V=-l(D^-d^). 

Write d=D- 2t, and we have F= 7rlt{D - 1\ 
20 = 7r^x0-5(4-0-5) = l-757r?, so that ?=3-638 inches. 



326 ELEMENTARY PRACTICAL MATHEMATICS 



(b) The areas are as the squaies, and the displacements are as the cubes 

of the lengths. The wetted area of i?= 12000 x 4 = 48000 sq. feet. 
The displacement of ^= 1500 x 8 = 12000. 

(c) If r is the radius of the pipe. (1) The area is ttv^ and the perimeter 

27rr, so that m=7rr^-^2Trr=^r. (2) The area is ^irr^, and the wetted 
perimeter is 7rr, so that 7n is still ^r. 

(d) As the common logarithm is 0-6314 x 0-43429 = 0-2742, the number 
is 1-880. 

3. The three parts (a), (b), and (c) must all be answered to get full 
marks : 

(a) If 0^7/" = a ; if .x is 5 when ?/ is 10, and if .r is 11 when y is 8, find n 
and or. What is the value of y when .r is 7 ? 

(6) The velocity of sound in air is 66-3 \/7 feet per second, where Hs the 
absolute temperature Centigrade, that is the ordinary temperature 
plus 273. What is the fractional change of velocity where the 
temperature alters from 10° C. to 15° C. ? 

(c) Assuming the earth to be a sphere of 8000 miles diameter, what is 
the circumference of the parallel of latitude 52° ? The earth makes 
one revolution in 24 hours (approximately) ; what is the speed at 
latitude 52° in miles per hour ? 

J7is. (a) As 5xl0" = llx8", we have log5 + w = log ll+wlog8 or 
0-6990 + n=l -041 4 + 0-9031 ti, 0-0969^1 = 03424, n = 3-5334. Also 
a=5xl03-5^= 17080, 0-8451+?? logy = 4-2324 or y = 9-093. 

(6) The velocities are proportional to \/283 and >/288 or 1 and >/288/283 
or 1 and 1*009. So that the fractional change of velocity is 0-009 
or 0-9 per cent. 

(c) Circumference 15474 miles, 644-75 miles per hour. 

4. There is a natural reservoir with irregular sides. When filled with 
water to the vertical height h feet above the lowest point, the following 
is the area A of the water surface in thousands of square feet : 



h 


« 


5 


10 


20 


30 


42 


50 


65 


75 


A 





220 


322 


435 


505 


560 586 


617 


624 



Find the average value of A between A = 10 and A = 65. 
What is A when A is 36 ? Find the volume of water which would raise 
the surface from A=35| to h = S6^. Ans. 520, 536, 536. 

5. The energy stored in similar fly-wheels is E—adPn^^ where d is the 
diameter and n the revolutions per minute ; a is a constant. A wheel 
whose diameter is 5 feet, revolving at 100 revolutions per minute, stores 
18,500 ft. -lb. ; find a. What is the diameter of a similar fly-wheel which 
will increase its store by 10,000 ft. -lb. when its speed increases from 149 
to 151 revolutions per minute ? 

Ans. a = 5-92xlO-^ Also, as 10000 = ac?fi(15l2- 1492) we have 



d^^ 



10000 
0-35514 



or c?= 7-761 feet. 



BOARD OF EDUCATION EXAMINATIONS 327 

6. There is a root of a^ + 5^— 11=0 between 1 and 2 ; find it, using 
squared paper, accurately to four significant figures. 

Writing jj=a:^ + bx-ll, for many values of x calculate y, plot on 
squared paper, and find for what value of ^, y is 0. 

We first try .^=1, and get 3/= -5 ; then x = 2, and 
get?^ = 7. 

It is therefore evident that our answer lies between 
these two values of x. 

Now try .r = l-5, and get ?/= -0-125, 

Plotting the three points now found on squared 
paper, I am induced to try .?;=1'52, and at once find 
that the answer is .*'=r511. . 

7. A steamer is moving at 20 feet per second towards the east ; the 
passengers notice that the smoke from the funnel streams off apparently 
towards the south-west with a speed of 10 feet per second ; what is the 
real speed of the wind and what is its direction ? If solved by actual 
drawing the work must be accurately done. 

Atis. The Avhole velocity of the smoke is the velocity of the vessel plus 
the velocity relatively to the vessel ; that is, drawing the vectors 
to scale, we find that the whole velocity of the smoke (that is, of 
the wind) is 14-73 feet per second in a direction from 28° 40' north 
of west or in a direction to 28° 40' south of east. 



X 


y 


1 

2 


-5 

7 


1-5 


-0-125 



8. If y = 20 + \/30+^'-^, take various values of x from 10 to 50 and 
calculate y. Plot on squared paper. What straight line agrees with 
the curve most nearly between these values ? Express it in the shape 
i/ = a + bx. Ans. y = 21 '25 + 0-973^. 

9. If the force which retards the falling of an object in a fluid is pro- 
portional to vs, where v is the velocity of falling and s is the area of the 
surface of the object, and if the force which accelerates falling is the 
weight of the object, show that as objects are smaller they fall more and 
more slowly. 

Becollect that of similar objects made of the same materials, the weights 
are as the cubes, the surfaces are as the squares of like dimensions. 

Ans. It is evident that the weight is proportional to s^ and the velocity 

of falling is such that s^ oc vs or v oc s^. Thus take spherical 
objects, if d is the diameter, s x dr, so that v cc d. 
Therefore objects of diameters 1, 0-1, 0-01, or 0*001 fall with velocities 
as 1 to 0-1 or 0-01 or O'OOl. 

10. A sliding piece is at the distance s feet from a point in its path at 
the time t seconds. Do not plot s and t. What is the average speed in 
each interval of time ? Assume that this is really the speed in the 
middle of the interval, and now plot time and speed on squared paper. 



8 


1-0000 


11054 


1-2146 


1-3268 


1-4432 


1-5624 


1-6857 


1-8118 


t 





0-1 


0-2 


0-3 


0-4 


0-5 


0/6 


0-7 



What is the approximate increase in speed between #=0-25 and 
0-35 ? What is approximately the acceleration when ^ = 0-3 ? 



328 ELEMENTARY PRACTICAL MATHEMATICS 

Ans. The average speeds in the interval are 



t 


Ss 


St 









1054 


01 






1092 


0-2 






1-122 


0-3 






1-164 


0-4 






1-192 


0-5 






1-233 


0-6 






1-261 


0-7 





The approximate increase in speed between t=0'25 
and ^=0*35 is from 1*122 to 1-164 feet per second in 
O'l second or 0"042 foot per second in O"! second, 
so that the acceleration at ^ = 0*3 is approximately 

^ =0-42 foot per second per second. 



11. The sections of the two ends of a barrel are each 12-25 sq. feet ; the 
middle section is 14-16 sq. feet ; the axial length of the barrel is 5 feet. 
What is its volume ? 

Simpson's Rule gives 1(12-35 + 1235 + 4 x 14-16) = 13*558 sq. feet as the 
average section, so that 13558 x 5 or 67*79 cubic feet is the volume. 

12. There is a machine consisting of two parts, whose weights are 
.randy. The cost of the machine in pounds is 12^* + 5y. The power of 
the machine is proportional to .Vj^. Find x and ?/ if the cost is £100, and 
if we desire to have the greatest power possible. Use squared paper if 
you please. 

Ans. 12.r+5y = 100, ?/ = 20-2-4r, .vi/ = 20x — 2'ix^, and we wish to find 
what value of x will make this a maximum. The calculus tells us 
that x=4},i/ = \0, gives the best result. But using squared paper, 
take the values of x, 3, 4, 4*5, 5, 6, etc., and calculate X2/. Plot xt/ 
and X, and we find the crest of the curve where x = 4l, and of 
course y = 20 — 2-4 x 4| = 10. 

13. According to a certain hypothesis the tensile stress in a rectangular 
cross-section of an iron hook at a distance y from a certain line through 
the centre of the section is proportional to 



P=- 



y-\-c 



1- 



R 



When jR = 10 and c = l, calculate p for various values of y from 3/ = 5 to 
y = - 5j and plot on squared paper. What is the average value of j3 ? 
For what value of y is the stress zero ? 

Ans. Evidently p = when y = - 1. Average /) = 2*55. 



BOARD OF EDUCATION EXAMINATIONS 329 



1910. Stage 3. 

1. The three parts (a), (6), and (c) must all be answered to get full 
marks : 

(a) Without using logarithms, compute by contracted methods, so that 
four significant figures shall be correct^ 

5-306 X 007632^7315. 
(6) Using logarithms, compute 

(22-15 -^4•139)-o•86. 

(c) The lengths of a degree of latitude and longitude in centimetres 
in latitude I are 
(1111-317 -5-688 cos 010* and (1 1 11 -1 64 cos ?- 0950 cos 3010*. 
The length of a sea mile (or 6082 feet) is 185380 cm. What are the 
lengths of a minute of latitude and of a minute of longitude in sea 
miles in the latitude 52" ? 

Ans, (a) 0-005536 ; (b) 0-2363 ; (c) 0-99600, 0-61687. 

2. A telephonic current of frequency pl^ir becomes of the value 

C= CqC-''^ sin (2)t - gx) 
in the distance of x miles, where 



V^VVO^^^)*^^ 



gives the value of h if the minus sign be taken, and the value of g if the 
plus sign be taken. When pljr is very large, what are the values of A 
and g approximately? If ^=0-05 x 10~<', r=88, and ^ = 5000, take two 
cases, (i) when ^ = and (ii) when ^ = 0*3, and in each case find the distance 
X in which the amplitude of C is halved. 



Ans. 



A = Wj, g=p\'kl; A'=6-609 and 38-58 miles. 



3. Find the value of cosh 0-1(1 +i\ where i means V — 1. 

Ans. 1+0-OU' or l[0°-573]. (See Art. 139.) 

4. To find the volume of part of a wedge, the frustum of a pyramid 
or of a cone, of part of a railway cutting or embankment, etc., we use the 
" Prismoidal Formula," which is " the sum of the areas of the end sections 
and four times the mid section, all divided by 6, is the average section ; 
this multiplied by the total length is the whole volume." Under what 
circumstances is this rule perfectly correct ? Prove its correctness. 

[The Prismoidal Formula is merely Simpson's Eule, and the question 
is fully answered in Art. 51.] 

5. If 2;=y + 2-i^, and if y is tabulated, find z approximately. 

ax 
Show both y and z as functions of x in curves : 



X 


4 


4-1 


4-2 


4-3 1 4-4 

1 


4-5 


y 


3-162 


3-548 


3-981 


4-467 


5-012 


5-623 



Ans. The values of z in the mid intervals, that is for .r=4-05, 4'15, etc., 
are 11-08, 12*42, 13-94, 15*64, 17-54. 



330 ELEMENTARY PRACTICAL MATHEMATICS 

6. A body capable of damped vibration is acted on by simply varying 
force which has a frequency /. If j; is the displacement of the body at 
any instant t, and if the motion is defined by 

we wish to study the forced vibration. 

Take a = l, 6=1*5, 71^ = 4 ; find ^ first when/=0-2547, and second when 
/= 0-3820. 

Ans. [Eefer to Art. 126.] Let q = 27rf, then x=smqtl(n^-q^-\-bqi), 
:r=sin qt{(4 — q^ + \ 'bqi). The values of q are 1-6 and 2-4. 
_ sin qt J _ ^^^ *Ji 

*^-r44 + 2^- ^""^ ^~-l•76 + 3•6^ 
in the two cases, 

or ^ = sin^^/2-799[59°-03] and ;r=sin^^/4-007[116°-05], 
or ^=0-3572 sin (g'jf- 59° -03) and ^=0-2496 sin (^jf-116°-05). 

7. The following values of x and y being given, tabulate ^- in each 

y . dx. Show in curves how the values 

of y, t^, and A depend upon x. 
dx 



X 


00 


0-1 


0-2 


0-3 


0-4 


0-5 


0-6 0-7 


0-8 


y 6-428 


7-071 


7-660 


8-192 


8-66 


9-063 


9-397 


9-659 


9-848 



Ans. For ^ and y . 8x for each interval, and also the values of A for 
ox 

a: =0-1, x=0% etc. 

1-89 
0-9754 

0-6750 i 1-4115 I 2-2041 I 30467 I 39329 I 4-8559 | 58087 I 67841 



6-43 


5-89 


5-32 


4-08 


4-03 


3-34 


2-62 


0-6750 


0-7365 


0-7926 


0-8426 


0-8862 


0-9230 


0-9528 



8. Here is a table giving values of y in terms of x^ and another 
giving values of u in terms of y. What is u when ^=83 ? 



X 


y 


y 


It 


7 
8 
9 


14-914 
16-128 
17-076 


15 
16 
17 


0-8169 
0-7118 
0-5543 



Atis. y is 16-44 and u is 0-6488. 

9. If pv=lOOt, and p = 3000 when ^ = 300, find v. If p = 30]0 and 
^ = 302, find the new v. If the second set of values be called 3000 + Sp, 
300 + 8?, and v + 8i\ what is 8v ? Now use the formula 

and calculate 8v in the new way. Why is there an error in the answer? 



BOARD OF EDUCATION EXAMINATIONS 331 

See Art. 143. v=]0, true Sy =0-033223. The new formula gives 
0'033333. The formula is true only when 8p and 8v are smaller and 
smaller without limit. 

10. The value of y, a periodic function of t, is here given for 12 equi- 
distant values of t covering the whole period. Express ;y in a Fourier 
Series. 

13-602, 18-468, 20-671, 20'182, 17*820, 14-346, 
10-130, 5-612, 1-877, 0-486, 2-500, 7-506. 

It ought not to be necessary to say that 18-468 is the second value. 

[See Art. 132 ; also Question 12, Stage 3, 1911.] 

A71S. y = 1 1 -1 + 10 sin ((^ + 10°) + sin {2cf) + 52°). 

11. To solve a^-20x-\-9=0 graphically, it is evident that we desire 
the value of x which will cause x^ to be equal to 20ji; — 9 ; plot therefore 
the curve y=x^^ and plot the straight line z = %)x — 9. Where they 
intersect we have the value of x desired. When the trial is made it will 
be found that there are three answers ; what are they ? 

Am. 4-23, 0-455, -4-68. 

12. On the indicator diagram of a gas engine the following are some 
readings of p pressure and v volume. The rate of reception of heat (if 
the gases are supposed to be receiving heat from an outside source and 
not from their own chemical action) is 

where k and K the important specifie heats are such that 

_L -24--^ 
K-k ^300* 



V 


20 


2-1 


2-2 


2-3 


2-4 


2-5 


2-6 


2-7 


2-8 


V 


84-5 


110 


176 


215 


231 


234 


2-26 i 213 


202 


V 


2-9 


3-0 


3-1 


3-2 


3-3 


3-4 


3-5 


3-6 


p 


192 


183 


175 


167 


159 i 

1 


152 


146 


140 



Find -^r- at three places; where v = 2-05, 3*55, and at the place of 
dv 
highest pressure. 

This question is answered in Ex. 89, Chap. XXVII. 

Ans. 1750, -115, 1160. 

13. When a shaft fails under the combined action of a bending 
moment M and a twisting moment 7\ according to what is called the 
internal friction hypothesis, 

ought to be constant where a is constant. Test if this is so, using the 
following numbers which have been published. Considerable errors in 
the observations must be expected. 



332 ELEMENTARY PRACTICAL MATHEMATICS 



3f 











1200 


1160 


1240 


2800 


2840 


T 


4320 


4360 


4308 


4338 


4326 


4368 


3836 


3846 


M 


2760 


4400 


4320 


4600 


5020 


5180 


5360 


T 


3804 


2416 


2438 


2060 





^ 






Ans. The student tabulates \IM^+T\ and plots this with M. Using 
the straight line which lies most evenly among the plotted points, 
he will find 5_oVi/2 + ^2 _ jf = 28830. 



INDEX. 



The references are to pages. 



Academic methods, vii. 
Acceleration, 70, 143, 299. 
Acceleration, angular, 63. 
Algebra, 25-50. 

Alternating electric currents, 218. 
Angles, 38, 41, 61-67, 78, 97, 264. 
Annuities, 37. 
Anti-logarithms, 76. 
Areas by integration, 151. 
Arithmetic, 1-24. 
Arithmetical progression, 35. 
Atmospheric friction, 23, 201. 
Atmospheric pressure, 191. 
Atmospheric resistance to projectiles, 

201. 
Average boy or man. vii, ix. 

Beams, 43, 44, 166, 169, 181. 
Belt, slipping of, 37, 117, 193. 
Binomial Theorem, 39. 
Brown's Relay, 285. 
Brown's Selective Circuit, 238. 

Calculus, infinitesimal, 69, 139. 
Centre of gravity, 93. 
Centrifugal acceleration, 299. 
Centrifugal pump or fan, 318. 
Chain, hanging, 177. 
Children at play, ix. 
Combined stresses, 322, 331, 
Commercial Arithmetic, 5. 
Compound Interest, 36, 38, 192. 
Compound Interest Law, 189-194. 
Conduction of heat, 274. 
Conductors, electrical, 48. 
Constants, useful, 73. 
Contracted methods, 2-4, 12. 
Co-ordinates, 102, 302-305. 
Cosh X, 269-273. 
Craigleith Quarry, 291. 
Crank, rotating, 214. 
Crank and connecting rod, 320. 



Curvature, 65, 166. 
I Curves y=ax'\ y = ae''\ 100, 126, 174. 
j Cycloid, 45, 105. 
I CyUnder, thick, 30. 

Damped vibrations, 44, 226. 
Demoivre's Theorem, 259, 266. 
Descriptive Geometry, 54. 
Differential calculus, 139. 
Differential equations, 194, 225, 247. 
Differentiation of a", 147, 150, 254. 

of sin X, 150. 

of e\ 150, 255. 
Differentiation, partial, 260. 
Discount, 36. 

Dishonesty in arithmetic, 1. 
dy/dzy 139. 

Eddy current loss, 49. 
Educational methods, defects of, ix. 
Effective current and voltage, 220. 
Electric condenser, 219. 
Electric vibrations, 230. 
Electrical conductors, 48. 
Electrical illustrations, 164, 189, 210, 
211, 218, 221, 222, 230, 236- 
241. 
EUipse, 104. 
Empirical formulae, 116, 129. 

misuse of, 130. 
Energy, kinetic, 163. 
Equations, solution of, 133. 

simple, 26. 

simultaneous, 26. 

quadratic, 26. 

roots of, 27, 29. 
Euchd, ix. 

Euler's theory of struts, 182. 
Examination Papers, 308-332. 
Experiments, bad and good, x. 
Exponential Theorem, 41, 87, 255. 
Extended Rules, 249-262. 



334 



INDEX 



Factors, 27. 

Fluids, flow of, 46, 186. 

rotating, 184. 
Forbes and Kelvin, 291. 
Forced vibrations, 228, 234. 
Formulae, empirical, misuse of, 130. 

simple for complex, 132. 

evaluation of, 20-24. 
Fourier's Series, 245, 323. 
Froude's Law, 33. 
Fundamental equations, 274-276. 

Gas engine, 43, 179. 
Gases, 260. 

flowing, 46, 171. 
Geometrical progression, 35. 
Geometry, syllabus, 51-53. 
Graphics, 216, 219. 
Gravity, centre of, 93. 
Guns, 31, 34. 

Half-time children, 82. 

Harmonic functions, 46. 

Heat conduction, 274, 290-293. 

Heat exercises, 41, 42. 

Heaviside, 280. 

History of Practical Mathematics, vii. 

Horse power of ships, 33. 

Hyperbola, 105. 

HyperboHc functions, 269. 

Ice, melting of, 165. 
Imaginary operators, 221, 268. 
Imaginary quantities, 194, 221, 

263-273. 
Impedance, 219. 
Induction coils, 49. 
Infinitesimal calculus, 139. 
Insurance office, 81. 
Integral calculus, 147. 
Interest, 6, 36, 38. 
Interpolation, 85. 

Kelvin, Lord, 189, 291. 
Kennelly, Professor, 288. 
Kinetic energy, 163. 

Latin, ix. 

Laws convertible to linear laws, 115. 

Lectures, viii. 

Level surface, 65. 

Linear Law, 106-125. 

Logarithmic paper, 121. 

Logarithms, 8-14, 74. 

Table of, 74. 

calculation of, 86. 

Maclaurin's Theorem, 259. 



Maxima and minima, 136-138, 168- 
173. 

Mensuration, 51-60, 88-98. 
Meridianal latitude, 50. 
Mid-ordinate Rule, 89-90. 
Multiplication in algebra, 299. 
Mutual induction, 241. 

Natural vibrations, 224-233. 
Networks of electric conductors, 239. 
Newton's Law of Cooling, 190. 

Odell's experiments, 209. 
Oil engine, 43. 

Operations, addition of, 217. 
Operator v/^, 221, 268. 
Operator, the, d/dt, 239. 

Parabola, area of, 152. 

Partial differentiation, 260. 

Pendulums, sympathetic, 241. 

Percentages, 4. 

Periodic functions in general, 244- 

248. 
Planimeter, 91. 
Population, 83. 
Position of a point, 102. 
Power, horse, of ships, 33. 
Practice, 6. 
Present value, 36. 
Pressure of atmosphere, 191. 
Prismoid, 89. 
Progressions, 35. 
Projectiles, 23, 201. 
Projection, 64. 
Proofs, 150, 249-262. 
Proportion, 7, 33. 

Reactance in electric circuits, 219. 
Restaurant, 113. 
Rings, 58, 59. 
Roget, Dr., 18. 
Rotating fluid, 184. 

Saucepans, 85. 

Series, 256-259. 

Seven Years' War, 102. 

Ships, resistance to, 33. 

Silk, price of, 80. 

Similar objects, 34. 

Simple formulae and complex, 132. 

Simple vibration, 211. 

Simpson's Rule, 88, 90. 

Sine function, 46, 211. 

Sines, curve of, 211. 

Sinh X, 269-273. 

SUde Rule, 15-18. 

Slope, 107, 139. 



INDEX 



Speed, 63, 68-71, 143. 

Squared paper, 80-86, 100-138. 

Steam, volume of, 165. 

Steam engine, 172. 

Straight line, 106, 139. 

Stresses, combined, 322, 331. 

Struts, 182. 

Subnormal, 175. 

Substitution of simple for complex 

formulae, 132. 
Subtangent, 175. 
Surface of revolution, 179. 
Suspension bridge, 177. 
Symbols, algebraic, 20-21. 

Tables of Logarithms, etc., 74-78. 

Tabulative differentiation, 154-161, 
202-3, 210. 
integration, 154-161. 

Tangent to curve, 174. 

Tanh x, 269-273. 

Taylor's Theorem, 257. 

Teachers, vii. 

Telegraph circuit, 275, 277-289. 

Telegraphy, wireless, 231-233, 241- 
243. 

Telephone circuits, 275, 277-289. 

Telephone transmitter, 315. 

Telephones, 47, 48. 

Text-books on Practical Mathe- 
matics, viii, xiii. 



Thermodynamics, 165, 179. 
Tides, 50. 

Train, speed of, 68, 142, 202. 
Trigonometry, 38, 41. 
Turbines, 24, 49, 198. 

Unreal quantities, 27, 194. 
Useful constants, 73. 

Variation, 33. 

Varley condenser, 285. 

Vectors, 294-307. 

addition of, 216, 294-298. 

differentiation of, 298, 321. 

notation for, 297. 

vector product of, 306. 

scalar product of, 300-306. 
Velocity, 68-71. 

angular, 63. 

of projectile, 71. 
Vertical line, 65. 
Vessels, resistance of, 33. 
Vibration, 46, 211. 

Water and ice, 165. 
and steam, 165. 
WiUans' Law, 117. 
Winding, best, of telephone or 

recorder, 286. 
WMtdker'a Almanack, 81. 
Wireless telegraphy, 231-233, 241-243. 



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